Answer:
6 billion years.
Step-by-step explanation:
According to the decay law, the amount of the radioactive substance that decays is proportional to each instant to the amount of substance present. Let [tex]P(t)[/tex] be the amount of [tex]^{235}U[/tex] and [tex]Q(t)[/tex] be the amount of [tex]^{238}U[/tex] after [tex]t[/tex] years.
Then, we obtain two differential equations
[tex]\frac{dP}{dt} = -k_1P \quad \frac{dQ}{dt} = -k_2Q[/tex]
where [tex]k_1[/tex] and [tex]k_2[/tex] are proportionality constants and the minus signs denotes decay.
Rearranging terms in the equations gives
[tex]\frac{dP}{P} = -k_1dt \quad \frac{dQ}{Q} = -k_2dt[/tex]
Now, the variables are separated, [tex]P[/tex] and [tex]Q[/tex] appear only on the left, and [tex]t[/tex] appears only on the right, so that we can integrate both sides.
[tex]\int \frac{dP}{P} = -k_1 \int dt \quad \int \frac{dQ}{Q} = -k_2\int dt[/tex]
which yields
[tex]\ln |P| = -k_1t + c_1 \quad \ln |Q| = -k_2t + c_2[/tex],
where [tex]c_1[/tex] and [tex]c_2[/tex] are constants of integration.
By taking exponents, we obtain
[tex]e^{\ln |P|} = e^{-k_1t + c_1} \quad e^{\ln |Q|} = e^{-k_12t + c_2}[/tex]
Hence,
[tex]P = C_1e^{-k_1t} \quad Q = C_2e^{-k_2t}[/tex],
where [tex]C_1 := \pm e^{c_1}[/tex] and [tex]C_2 := \pm e^{c_2}[/tex].
Since the amounts of the uranium isotopes were the same initially, we obtain the initial condition
[tex]P(0) = Q(0) = C[/tex]
Substituting 0 for [tex]P[/tex] in the general solution gives
[tex]C = P(0) = C_1 e^0 \implies C= C_1[/tex]
Similarly, we obtain [tex]C = C_2[/tex] and
[tex]P = Ce^{-k_1t} \quad Q = Ce^{-k_2t}[/tex]
The relation between the decay constant [tex]k[/tex] and the half-life is given by
[tex]\tau = \frac{\ln 2}{k}[/tex]
We can use this fact to determine the numeric values of the decay constants [tex]k_1[/tex] and [tex]k_2[/tex]. Thus,
[tex]4.51 \times 10^9 = \frac{\ln 2}{k_1} \implies k_1 = \frac{\ln 2}{4.51 \times 10^9}[/tex]
and
[tex]7.10 \times 10^8 = \frac{\ln 2}{k_2} \implies k_2 = \frac{\ln 2}{7.10 \times 10^8}[/tex]
Therefore,
[tex]P = Ce^{-\frac{\ln 2}{4.51 \times 10^9}t} \quad Q = Ce^{-k_2 = \frac{\ln 2}{7.10 \times 10^8}t}[/tex]
We have that
[tex]\frac{P(t)}{Q(t)} = 137.7[/tex]
Hence,
[tex]\frac{Ce^{-\frac{\ln 2}{4.51 \times 10^9}t} }{Ce^{-k_2 = \frac{\ln 2}{7.10 \times 10^8}t}} = 137.7[/tex]
Solving for [tex]t[/tex] yields [tex]t \approx 6 \times 10^9[/tex], which means that the age of the universe is about 6 billion years.
The age of the universe, based on the given ratio of 238U to 235U isotopes and their half-lives, is approximately 8750 years.
To calculate the age of the universe based on the ratio of 238U to 235U isotopes, we can use the concept of radioactive decay and the given half-lives.
The ratio of 238U to 235U is currently 137.7 to 1. This means that over time, 238U has been decaying into other elements, while 235U has been decaying into different elements at different rates due to their distinct half-lives.
We'll start by calculating the number of half-lives that have passed for each isotope to reach the current ratio:
For 238U:
(Number of half-lives) = (Age of the universe) / (Half-life of 238U)
(Number of half-lives) = (Age of the universe) / (4.51 × [tex]10^9[/tex] years)
For 235U:
(Number of half-lives) = (Age of the universe) / (Half-life of 235U)
(Number of half-lives) = (Age of the universe) / (7.10 × [tex]10^8[/tex] years)
Since there is a ratio of 137.7 to 1, it means that the number of half-lives for 238U should be 137.7 times that of 235U:
(Number of half-lives for 238U) = 137.7 × (Number of half-lives for 235U)
Now, we can set up an equation using these relationships:
(137.7) × [(Age of the universe) / (4.51 × [tex]10^9[/tex] years)] = (Age of the universe) / (7.10 × 1[tex]0^8[/tex]years)
Now, we can solve for the "Age of the universe":
137.7 × (4.51 × [tex]10^9[/tex]) = 7.10 × [tex]10^8[/tex] × (Age of the universe)
(Age of the universe) = (137.7 × 4.51 × [tex]10^9[/tex]) / (7.10 × [tex]10^8[/tex])
(Age of the universe) ≈ 8750 years
So, according to this cosmological theory, the age of the universe is approximately 8750 years.
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Find the missing lengths:
OK=1 and OL=3, find KH and LH.
Answer:
KH = 2 and LH = 2√3
Step-by-step explanation:
Using Euclidean theorem for the right triangle.
∵ ΔLHK is a right triangle at H
OK = 1 , OL = 3
KL = KO + OL = 1 + 3 = 4
KH² = KO * KL = 1 * 4 = 4
KH = √4 = 2
And LH² = LO * LK = 3 * 4 = 12
∴ LH = √12 = 2√3
Find the area of the parallelogram with vertices Ps1, 0, 2d, Qs3, 3, 3d, Rs7, 5, 8d, and Ss5, 2, 7d.
To find the area of a parallelogram, you can use the cross product of two vectors created from the vertices. The magnitude of the resulting vector represents the area of the parallelogram. Ensure to put an absolute value on the final result.
Explanation:The subject of this question is geometry, specifically finding the area of a parallelogram given the vertices. You can calculate the area of a parallelogram using the cross product of two vectors. First, create two vectors from the given vertices, for example, PQ = Q - P and PR = R - P. For your question, let's take P as s1, 0, 2d, Q as s3, 3, 3d, and R as s7, 5, 8d. So, PQ = s2, 3, 1d and PR = s6, 5, 6d. The cross product of these two vectors would give a vector perpendicular to both, whose magnitude represents the area of the parallelogram spanned by the vectors PQ and PR. The magnitude (or length) of a vector ABC = sA, B, Cd is calculated as √(A² + B² + C²).
Using these formulas and calculations, you should be able to find the area of your parallelogram. Remember to put absolute value on the final answer as area cannot be negative.
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If Ms. P wants to withdraw $900 from an account earning 4% average annual interest rate at the start of each year for 7 years, how much must she have in the account today?
Answer:
Amount he must have in his account today is $5,617.92
Step-by-step explanation:
Data provided in the question:
Regular withdraw amount = $900
Average annual interest rate, i = 4% = 0.04
Time, n = 7 years
Now,
Present Value = [tex]C \times\left[ \frac{1-(1+i)^{-n}}{i} \right] \times(1 + i)[/tex]
here,
C = Regular withdraw amount
Thus,
Present Value = [tex]C \times\left[ \frac{1-(1+i)^{-n}}{i} \right] \times(1 + i)[/tex]
Present Value = [tex]900 \times\left[ \frac{1-(1+0.04)^{-7}}{ 0.04 } \right] \times(1 + 0.04)[/tex]
Present Value = [tex]936 \times\left[ \frac{1 - 1.04^{-7}}{ 0.04} \right][/tex]
Present Value = [tex]936 \times\left[ \frac{1 - 0.759918}{ 0.04} \right][/tex]
Present Value = 936 × 6.00205
or
Present Value = $5,617.92
Hence,
Amount he must have in his account today is $5,617.92
Geno wants to purchase gym membership. He has no more than y dollars to spend. Total Fitness charges an initial fee of $100 plus $30 per month. Gymania charges initial fee of $25 plus $50 per month. Write a system of equations that can be used to determine which company offers the better deal.
Answer:
Gymania is a better deal if the membership is for 3 months and below.
Total Fitness is a better if the membership is for 4 months and above.
Step-by-step explanation:
Let the number of months be 'x'.
Given:
Money Geno has = 'y' dollars.
Total Fitness charges:
Monthly fee = $30
Initial fee = $100
Gymania charges:
Monthly fee = $50
Initial fee = $25
Total charges is equal to the sum of initial fee and monthly fee multiplied by number of months.
So, for 'x' months, monthly fee charged by Total Fitness = [tex]30x[/tex]
For 'x' months, monthly fee charged by Gymania = [tex]50x[/tex]
Now, total charge by Total Fitness = Initial fee + Fee for 'x' months
Total charge by Total Fitness = [tex]100+30x[/tex]
Now, total charge by Gymania = Initial fee + Fee for 'x' months
Total charge by Gymania = [tex]25+50x[/tex]
Now, Geno has only 'y' dollars to spend. So, 'y' must be less than or equal to the total charge.
Therefore, the total charge for each membership is:
[tex]y=30x+100\\\\y= 50x+25[/tex]
Now, we graph both the equations. The graph is shown below.
From the graph, it is clear that, the total cost for Gymania (blue line) is less than that of Total Fitness (red line) till number of months equals 3.75 or 3 months. After 3.75 months, the graph of Gymania is above Total Fitness. So, if the membership is 4 months or above, then Total Fitness is more efficient.
Therefore, Gymania is a better deal if the membership is for 3 months and below.
Total Fitness is a better if the membership is for 4 months and above.
To determine which company offers the better deal for a gym membership, set up a system of equations for the two companies' costs and compare. Total Fitness charges $30 per month, while Gymania charges $50 per month. The equation is solved to find the break-even point where their costs are equal.
Explanation:To determine which company offers the better deal, we can set up a system of equations based on the given information:
Let x be the total number of months for the gym membership.
Total Fitness charges an initial fee of $100 plus $30 per month, so the total cost can be represented by the equation: y = 30x + 100.
Gymania charges an initial fee of $25 plus $50 per month, so the total cost can be represented by the equation: y = 50x + 25.
To compare the two deals, we need to find the values of x where the total cost is the same for both companies. We can set up the following equation:
30x + 100 = 50x + 25.
Simplifying, we get:
20x = 75.
Dividing both sides by 20, we find that x = 3.75.
Since x represents the number of months, it cannot be a decimal, so we round up to the nearest whole number. Therefore, Geno should join Total Fitness if he plans to have the membership for 4 or more months, and Gymania if he plans to have the membership for 3 or fewer months.
Find the exact value of cos theta, given that sin thetaequalsStartFraction 15 Over 17 EndFraction and theta is in quadrant II. Rationalize denominators when applicable.
Answer:
[tex] cos \theta = -\frac{8}{17}[/tex]
Step-by-step explanation:
For this case we know that:
[tex] sin \theta = \frac{15}{17}[/tex]
And we want to find the value for [tex] cos \theta[/tex], so then we can use the following basic identity:
[tex] cos^2 \theta + sin^2 \theta =1 [/tex]
And if we solve for [tex] cos \theta [/tex] we got:
[tex] cos^2 \theta = 1- sin^2 \theta[/tex]
[tex] cos \theta =\pm \sqrt{1-sin^2 \theta}[/tex]
And if we replace the value given we got:
[tex] cos \theta =\pm \sqrt{1- (\frac{15}{17})^2}=\sqrt{\frac{64}{289}}=\frac{\sqrt{64}}{\sqrt{289}}=\frac{8}{17}[/tex]
For our case we know that the angle is on the II quadrant, and on this quadrant we know that the sine is positive but the cosine is negative so then the correct answer for this case would be:
[tex] cos \theta = -\frac{8}{17}[/tex]
Answer:
It is D
Step-by-step explanation:
EDGE 2021
Solve the triangle
Solve sides
Answer:
Step-by-step explanation:
Triangle RST is a right angle triangle.
From the given right angle triangle
RT represents the hypotenuse of the right angle triangle.
With 26 degrees as the reference angle,
ST represents the adjacent side of the right angle triangle.
RS represents the opposite side of the right angle triangle.
1) To determine RS, we would apply trigonometric ratio
Sin θ = opposite side/hypotenuse Therefore,
Sin 26 = RS/9.1
RS = 9.1Sin26 = 9.1 × 0.4384
RS = 4.0
2) To determine ST, we would apply trigonometric ratio
Cos θ = adjacent side/hypotenuse Therefore,
Cos 26 = ST/9.1
ST = 9.1Cos26 = 9.1 × 0.8988
ST = 8.1
3) The sum of the angles in a triangle is 180 degrees. Therefore,
∠R + 26 + 90 = 180
∠R = 180 - (26 + 90)
∠R = 64 degrees
Carlos will buy coffee and hit chocolate for his co-workers. Each cup of coffee costs $2.25 and each cup of hit chocolate costs $1.50. If he pays a total of $15.75 for 8 cups, how many of each did he buy?
Answer: he bought 5 cups of coffee.
He bought 3 cups of hit chocolate
Step-by-step explanation:
Let x represent the number cups of coffee that he bought.
Let y represent the the number cups of of hit chocolate that he bought.
He bought a total of 8 cups coffee and hit chocolate. This means that
x + y = 8
Each cup of coffee costs $2.25 and each cup of hit chocolate costs $1.50. If he pays a total of $15.75 for 8 cups, it means that
2.25x + 1.5y = 15.75 - - - - - - - - - - - - 1
Substituting x = 8 - y into equation 1, it becomes
2.25(8 - y) + 1.5y = 15.75
18 - 2.25y + 1.5y = 15.75
- 2.25y + 1.5y = 15.75 - 18
- 0.75y = - 2.25
y = - 2.25/- 0.75
y = 3
Substituting y = 3 into x = 8 - y, it becomes
x = 8 - 3 = 5
Write the ratios for sine C cosine C and tangent C
Answer:
sin C = 7/27, cos C = 24/27 and tan C = 7/24
Step-by-step explanation:
sin C = opposite/hypotenuse
= 7/27
cos C = adjacent/hypotenuse
= 24/27
tan C = opposite/ adjacent
= 7/24
Answer:
sin c - 25/7
cos c - 7/24
tan c - 25/24
Step-by-step explanation:
A car manufacturer wants to assess customer satisfaction for cars sold during the previous year.(a) Describe the population involved.(b) Is the population involved hypothetical or not?
Answer:
a) The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people
b) For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.
Step-by-step explanation:
Part a
The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people
Part b
For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.
a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.
b. The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.
Data handlingthe process of ensuring that research data is stored.
How to take data?a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.
b. The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.
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In a 2-sample z-test for two proportions, you find the following: sub(hat(p),1)
Answer:
[tex]z=\frac{0.32-0.36}{\sqrt{0.34(1-0.34)(\frac{1}{50}+\frac{1}{50})}}=-0.422[/tex]
So on this case the only option that satisfy the calculated statistic is:
z=-0.42
Step-by-step explanation:
Assuming this complete problem: "In a 2-sample z-test for two proportions, you find the following:
^P1 = 0.32, (n,1)=50
^P,2= 0.36, (n,2)=50
Find the test statistic you will use while executing this test:
z=-0.67 , z=±1.64 , z=-1.96 , z=0.34 , z=-0.42"
Solution to the problem
Data given and notation
[tex]n_{1}=50[/tex] sample 1 selected
[tex]n_{2}=50[/tex] sample 2 selected
[tex]p_{1}=0.32[/tex] represent the sample proportion for 1
[tex]p_{2}=0.36[/tex] represent the sample proportion for 2
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex] (1)
Or equivalently:
[tex]z=\frac{\hat p_{2}-\hat p_{1}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{1}+X_{1}}{n_{1}+n_{1}}=\frac{\hat p_1 +\hat p_2}{2}=\frac{0.32+0.36}{2}=0.34[/tex]
Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.32-0.36}{\sqrt{0.34(1-0.34)(\frac{1}{50}+\frac{1}{50})}}=-0.422[/tex]
So on this case the only option that satisfy the calculated statistic is:
z=-0.42
In analyzing the recordings, what would you define as the unit for analysis? Why? How many data units (in rough estimates) are you likely to get based on this decision? Does the estimated number of data units seem adequate? Why or why not?
Answer:
a) The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.
b) They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians
c) For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.
Step-by-step explanation:
Assuming this complete question: "Suppose you are interested in the behaviors of physicians that have high ratings of patient satisfaction. The research goal is to identify the behaviors in the natural clinical settings of these successful physicians so that these behaviors can be built into the curricula of medical preparation programs. The main data were collected by the video recording of five randomly selected physician-patient encounters from 10 physicians who have been rated highly by patients in a reliable satisfaction survey. In analyzing the recordings, what would you define as the unit for analysis? Why? How many data units (in rough estimates) are you likely to get based on this decision? Does the estimated number of data units seem adequate? Why or why not? "
In analyzing the recordings, what would you define as the unit for analysis? Why?
The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.
How many data units (in rough estimates) are you likely to get based on this decision?
They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians
Does the estimated number of data units seem adequate? Why or why not?
For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.
A chemist needs to mix a 39% salt solution with a 70% salt solution to make 40 liters of a 55% salt solution. How many liters of each solution should the chemist mix to get the desired result?
Solution with 39% salt:
.
Solution with 70% salt:
Thanks a ton!! :)
Step-by-step explanation:
If x is the volume of 39% solution, and y is the volume of 70% solution, then:
x + y = 40
0.39x + 0.70y = 0.55(40)
Solve the system of equations.
0.39x + 0.70(40 − x) = 0.55(40)
0.39x + 28 − 0.70x = 22
6 = 0.31x
x = 19.4
y = 20.6
The chemist needs 19.4 liters of 39% solution and 20.6 liters of 70% solution.
Fereydoun is conducting a study of the annual incomes of high school teachers in metropolitan areas of fewer than 100,000 population, and in metropolitan areas having greater than 500,000 population.
If computed z value is 16.1, can he conclude that the annual incomes of high school teachers in metropolitan areas having greater than 500,000 population are significantly greater than those paid in areas with fewer than 100,000 population, at 0.05 level of significance?
Answer:
Yes, the claim can be concluded.
Step-by-step explanation:
We are given the following in the question:
Alpha, α = 0.05
The null hypothesis and alternate hypothesis can be designed in the following manner:
[tex]H_{0}: \mu_{500,000} = \mu_{100,000}\\H_A: \mu_{500,000} > \mu_{100,000}[/tex]
This is a one tailed(right) test.
[tex]z_{stat} = 16.2[/tex]
Now, we calculate the p - value from standard table.
P-value = 0.00001
Since the p value is less than the significance level, we fail to accept the null hypothesis and reject it.
We accept the alternate hypothesis.
Thus, we conclude that there is enough evidence to support the claim that the annual incomes of high school teachers in metropolitan areas having greater than 500,000 population are significantly greater than those paid in areas with fewer than 100,000 population.
Determine for which values of m the function variant Φ(x) = x^m is a solution to the given equation. a. 3x^2 (d^2y/dx^2) + 11x(dy/dx) - 3y = 0 b. x^2 (d^2y/dx^2) - x(dy/dx) - 5y = 0
Answer:
a) m = -9 or m = 1
b) m = 1 + √6 or m = 1 -√6
Step-by-step explanation:
for
Φ(x) = x^m
then
dΦ/dx (x) = m*x^(m-1)
d²Φ/dx² (x) = m*(m-1)*x^(m-2)
then
for a
3x^2 (d^2y/dx^2) + 11x(dy/dx) - 3y = 0
3x^2*m*(m-1)*x^(m-2) + 11*x* m*x^(m-1) - 3*x^m = 0
3*m*(m-1)*x^m + 11*m*x^m- 3*x^m = 0
dividing by x^m
3*m*(m-1) + 11*m - 3 =0
3*m² + 8 m - 3 =0
m= [-8 ± √(64 + 4*3*3)]/2 = (-8±10)/2
m₁ = -9 , m₂= 1
then Φ(x) = x^m is a solution for the equation a , when m = -9 or m = 1
for b)
x^2 (d^2y/dx^2) - x(dy/dx) - 5y = 0
x^2*m*(m-1)*x^(m-2) - x* m*x^(m-1) - 5*x^m = 0
m*(m-1)*x^m -m *x^m- 5*x^m = 0
dividing by x^m
m*(m-1) -m - 5 =0
m² - 2 m - 5 =0
m= [2 ± √(4 + 4*1*5)]/2 = (2±√24)/2 = 1 ±√6
m₁ = 1 + √6 , m₂ = 1 - √6
then Φ(x) = x^m is a solution for the equation b , when m = 1 + √6 or m = 1 - √6
Answer
a) m = -3 or 1/3
b) m = 1 + root 6 or 1 - root 6
Step-by-step explanation:
The step by step calculation is as shown in the attachment.
Let X denote the size of a bodily injury claim and Y denote the size of the corresponding property damage claim. Let Z1 = X + Y. From prior experience we know Var(X) = 144, Var(Y) = 64 and Var(X + Y) = 308. It is expected that bodily injury claims will rise 10% next year and property damage will rise by a fixed amount of 5. Let Z2 be the new trial of bodily injury and property damage. Compute Cov(Z1, Z2 ).
Final answer:
To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y individually. Using the given variances and calculations, we can find Cov(Z1, Z2) = 308 + 0.1(144 + 2Cov(X, Y) + 64).
Explanation:
To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y first.
Cov(X, X): Since Var(X) is given as 144, Cov(X, X) = Var(X) = 144Cov(X, Y): Cov(X, Y) = Cov(Y, X) because covariance is commutative. Also, Cov(X, Y) = Cov(Z1 - Y, Y) = Cov(Z1, Y) - Cov(Y, Y) = Cov(X + Y, Y) - Var(Y) = Cov(Z1, Y) - Var(Y) = Cov(Z1, Y) - 64Cov(Y, X): Since Cov(Y, X) = Cov(X, Y), we can use Cov(X, Y) from the previous step.Cov(Y, Y): Cov(Y, Y) = Var(Y) = 64Now, we can calculate Cov(Z1, Z2) using the following formula:
Cov(Z1, Z2) = Cov(X + Y, X + 0.1X + 5) = Cov(Z1, Z1 + 0.1X + 5) = Cov(Z1, Z1) + Cov(Z1, 0.1X) + Cov(Z1, 5) = Var(Z1) + 0.1Cov(Z1, X) + 0 = Var(X + Y) + 0.1Cov(Z1, X) + 0 = 308 + 0.1(Cov(X, X) + Cov(X, Y) + Cov(Y, X) + Cov(Y, Y)) + 0 = 308 + 0.1(144 + Cov(X, Y) + Cov(Y, X) + 64) + 0 = 308 + 0.1(144 + 2Cov(X, Y) + 64)
What are the hypotheses for testing to see if a correlation is statistically significant?
Answer:
Step-by-step explanation:
For testing of significance of correlation coefficient denoted by r, we create hypotheses in three ways
They are one tailed, two tailed. One tailed can be stated as right tailed and also left tailed.
The null hypothesis would normally be as r=0
Verbally we can say this there is no association between the dependent and independent variable (linear)
Against this alternate hypothesis is created as
either r not equal to 0
or r>0 or r<0
If r not equal to 0, we say two tailed hypothesis test
If r>0 is alternate hypothesis, it is right tailed test
If r<0 is alternate hypothesis, then it is left tailed test.
55. If a test statistic falls in the critical region the null hypothesis is _______________. a) Rejected b) Not Rejected c) It depends d) accepted
Answer:
Option a) Rejected
Step-by-step explanation:
We define critical region as:
It is also known as the rejection region.It is the region where the null hypothesis is rejected.Thus, if our calculated test statistic lies in the critical region, we fail to accept the null hypothesis and reject it.For right tailed test: The critical region is the values greater than the critical value.For left tailed test: The critical region is the values less than the critical value.The critical region for two tailed test is the value apart from the range of critical values.Thus,
If a test statistic falls in the critical region the null hypothesis is rejected.
The number of major earthquakes in a year is approximately normally distributed with a mean of 20.8 and a standard deviation of 4.5. a) Find the probability that in a given year there will be less than 21 earthquakes. b) Find the probability that in a given year there will be between 18 and 23 earthquakes.
Answer:
a) 51.60% probability that in a given year there will be less than 21 earthquakes.
b) 49.35% probability that in a given year there will be between 18 and 23 earthquakes.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 20.8, \sigma = 4.5[/tex]
a) Find the probability that in a given year there will be less than 21 earthquakes.
This is the pvalue of Z when X = 21. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{21 - 20.8}{4.5}[/tex]
[tex]Z = 0.04[/tex]
[tex]Z = 0.04[/tex] has a pvalue of 0.5160.
So there is a 51.60% probability that in a given year there will be less than 21 earthquakes.
b) Find the probability that in a given year there will be between 18 and 23 earthquakes.
This is the pvalue of Z when X = 23 subtracted by the pvalue of Z when X = 18. So:
X = 23
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{23 - 20.8}{4.5}[/tex]
[tex]Z = 0.71[/tex]
[tex]Z = 0.71[/tex] has a pvalue of 0.7611
X = 18
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{18 - 20.8}{4.5}[/tex]
[tex]Z = -0.62[/tex]
[tex]Z = -0.62[/tex] has a pvalue of 0.2676
So there is a 0.7611 - 0.2676 = 0.4935 = 49.35% probability that in a given year there will be between 18 and 23 earthquakes.
Final answer:
To find the probability of specific numbers of earthquakes occurring within a year using a normal distribution, one calculates the Z-scores for those numbers and looks up or calculates the corresponding probabilities.
Explanation:
The question involves finding probabilities related to the number of earthquakes in a year, which is modeled using a normal distribution. To find these probabilities, we'll use the mean μ = 20.8 and the standard deviation σ = 4.5 of the distribution.
a) Probability of less than 21 earthquakes
To find the probability of there being less than 21 earthquakes in a year, we calculate the Z-score for 21:
Z = (X - μ) / σ = (21 - 20.8) / 4.5 ≈ 0.04
Looking up this Z-score on a standard normal distribution table or using a calculator, we find the corresponding probability and note that it's slightly more than 0.5, indicating a little over a 50% chance.
b) Probability of 18 to 23 earthquakes
Finding the Z-scores for 18 and 23:
Z for 18 = (18 - 20.8) / 4.5 ≈ -0.62
Z for 23 = (23 - 20.8) / 4.5 ≈ 0.49
You then look up these Z-scores on a standard normal distribution table to find the probabilities for each and subtract the smaller from the larger to get the probability of having between 18 and 23 earthquakes in a year. This method shows that there's a significant chance, typically around 40% to 50%, though the specific value requires precise Z-score to probability conversion.
Eight less than the product of a number n and 1/5 is no more than 96
Final answer:
The question translates to the inequality (1/5)n - 8 ≤ 96, which is solved by adding 8 to both sides and then multiplying by 5, resulting in n ≤ 520.
Explanation:
The question involves translating a word problem into a mathematical inequality. The phrase 'Eight less than the product of a number n and 1/5' can be written as (1/5)n - 8. When it states that this is 'no more than 96', it implies that the expression should be less than or equal to 96. Therefore, the inequality we need to solve is (1/5)n - 8 ≤ 96.
Now, let's solve this inequality step-by-step:
Add 8 to both sides of the inequality: (1/5)n ≤ 104.
Multiply both sides by 5 to solve for n: n ≤ 520.
This gives us the solution to the inequality, indicating that the number n can be any value less than or equal to 520 to satisfy the initial condition.
Vehicles arrive at an intersection at a rate of 400 veh/h according to a Poisson distribution. What is the probability that more than five vehicles will arrive in a one-minute interval?
Answer:
0.6547 or 65.47%
Step-by-step explanation:
One minute equals 1/60 of an hour, the mean number of occurrences in that interval is:
[tex]\lambda =\frac{400}{60}=6.6667[/tex]
The poisson distribution is described by the following equation:
[tex]P(x) =\frac{\lambda^{x}*e^{-\lambda}}{x!}[/tex]
The probability that more than 5 vehicles will arrive is:
[tex]P(x>5)= 1-(P(0)+P(1)+P(2)+P(3)+P(4)+P(5))\\P(x>5) = 1-(\frac{6.667^{0}*e^{-6.667}}{1}+\frac{6.667^{1}*e^{-6.667}}{1}+\frac{6.667^{2}*e^{-6.667}}{2}+\frac{6.667^{3}*e^{-6.667}}{3*2}+\frac{6.667^{4}*e^{-6.667}}{4*3*2}+\frac{6.667^{5}*e^{-6.667}}{5*4*3*2})\\P(x>5)=1-(0.00127+0.00848+0.02827+ 0.06283+0.10473+0.13965)\\P(x>5)=0.6547[/tex]
The probability that more than five vehicles will arrive in a one-minute interval is 0.6547 or 65.47%.
The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.
Step 1
Given that vehicles arrive at an intersection at a rate of 400 vehicles per hour, and this follows a Poisson distribution, we want to find the probability that more than five vehicles will arrive in a one-minute interval.
First, convert the arrival rate to a one-minute interval. Since there are 60 minutes in an hour, the arrival rate per minute is:
[tex]\[ \lambda = \frac{400 \, \text{veh/h}}{60} = \frac{400}{60} \approx 6.67 \, \text{veh/min} \][/tex]
The Poisson distribution formula for the probability of observing k events in an interval is:
[tex]\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \][/tex]
We need the probability that more than five vehicles arrive in one minute:
[tex]\[ P(X > 5) = 1 - P(X \leq 5) \][/tex]
Step 2
First, calculate [tex]\( P(X \leq 5) \)[/tex] by summing the probabilities for k = 0 to k = 5 :
[tex]\[ P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-6.67} 6.67^k}{k!} \][/tex]
Let's calculate these probabilities:
- For k = 0 :
[tex]\[ P(X = 0) = \frac{e^{-6.67} 6.67^0}{0!} = e^{-6.67} \][/tex]
- For k = 1 :
[tex]\[ P(X = 1) = \frac{e^{-6.67} 6.67^1}{1!} = e^{-6.67} \times 6.67 \][/tex]
- For k = 2 :
[tex]\[ P(X = 2) = \frac{e^{-6.67} 6.67^2}{2!} = e^{-6.67} \times \frac{6.67^2}{2} \][/tex]
- For k = 3:
[tex]\[ P(X = 3) = \frac{e^{-6.67} 6.67^3}{3!} = e^{-6.67} \times \frac{6.67^3}{6} \][/tex]
- For k = 4 :
[tex]\[ P(X = 4) = \frac{e^{-6.67} 6.67^4}{4!} = e^{-6.67} \times \frac{6.67^4}{24} \][/tex]
- For k = 5 :
[tex]\[ P(X = 5) = \frac{e^{-6.67} 6.67^5}{5!} = e^{-6.67} \times \frac{6.67^5}{120} \][/tex]
Sum these probabilities to find [tex]\( P(X \leq 5) \)[/tex].
Step 3
Next, we calculate [tex]\( e^{-6.67} \)[/tex] and the terms:
[tex]\[e^{-6.67} \approx 0.00126\][/tex]
[tex]\[P(X = 0) \approx 0.00126\][/tex]
[tex]\[P(X = 1) \approx 0.00126 \times 6.67 = 0.0084\][/tex]
[tex]\[P(X = 2) \approx 0.00126 \times \frac{6.67^2}{2} = 0.0280\][/tex]
[tex]\[P(X = 3) \approx 0.00126 \times \frac{6.67^3}{6} = 0.0622\][/tex]
[tex]\[P(X = 4) \approx 0.00126 \times \frac{6.67^4}{24} = 0.1037\][/tex]
[tex]\[P(X = 5) \approx 0.00126 \times \frac{6.67^5}{120} = 0.1382\][/tex]
Sum these probabilities:
[tex]\[P(X \leq 5) \approx 0.00126 + 0.0084 + 0.0280 + 0.0622 + 0.1037 + 0.1382 = 0.34176\][/tex]
Finally, the probability that more than five vehicles will arrive in a one-minute interval is:
[tex]\[P(X > 5) = 1 - P(X \leq 5) = 1 - 0.34176 = 0.65824\][/tex]
The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.
A population consists of the following N = 5 scores: 0, 6, 4, 3, and 12.
(a) Compute µ and σ for the population.
(b) Find the z-score for each score in the population.
(c) Transform the original population into a new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20.
Answer:a) μ = 5 and σ = 16
b) z-score are -0.3125, 0.0625, -0.0625, -0.125, 0.4375
c) New population of N=5 scores are 93.75, 101.25, 98.75, 97.5, 108.75
Step-by-step explanation:
The detailed explanation can be found in the attached pictures
The new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135
(a) Compute µ and σ for the population.
The dataset is given as:
0, 6, 4, 3, and 12.
The mean is calculated as:
[tex]\mu = \frac{\sum x}n[/tex]
So, we have:
[tex]\mu = \frac{0 + 6 + 4 + 3 + 12}5[/tex]
[tex]\mu = 5[/tex]
The standard deviation is calculated as:
[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)^2}n}[/tex]
This gives
[tex]\sigma = \sqrt{\frac{(0 - 5)^2 + (6- 5)^2 + (4- 5)^2 + (3- 5)^2 + (12- 5)^2}5[/tex]
[tex]\sigma = \sqrt{\frac{80}5[/tex]
[tex]\sigma = \sqrt{16[/tex]
[tex]\sigma = 4[/tex]
Hence, the values of μ and σ are μ = 5 and σ = 4
(b) The z-scores
This is calculated as:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
When x = 0, 6, 4, 3, and 12.
We have:
[tex]z = \frac{0 - 5}{4} = 1.25[/tex]
[tex]z = \frac{6 - 5}{4} = 0.25[/tex]
[tex]z = \frac{4 - 5}{4} = -0.25[/tex]
[tex]z = \frac{3 - 5}{4} = -0.5[/tex]
[tex]z = \frac{12 - 5}{4} = 1.75[/tex]
Hence, the z-scores are 1.25, 0.25, -0.25, -0.5 and 1.75
(c) Transform the new population
We have:
N = 5, µ = 100 and σ = 20.
In (b), we have:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
Make x the subject
[tex]x = \mu + z\sigma[/tex]
This gives
[tex]x_i = \mu + z_i\sigma[/tex]
So, we have:
[tex]x_1 = 100 + 1.25* 20 = 125[/tex]
[tex]x_2 = 100 + 0.25* 20 = 105[/tex]
[tex]x_3 = 100 - 0.25* 20 = 95[/tex]
[tex]x_4 = 100 - 0.5* 20 = 90[/tex]
[tex]x_5 = 100 + 1.75* 20 = 135[/tex]
Hence, the new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135
Read more about z-scores at:
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The doctor has told Cal O'Ree that during his ten weeks of working out at the gym, he can expect each week's weight loss to be $1\%$ of his weight at the end of the previous week. His weight at the beginning of the workouts is $244$ pounds. How many pounds does he expect to weigh at the end of the ten weeks?
Cal's expected weight after 10 weeks can be calculated using the exponential decay formula, with his initial weight being 244 lbs, the decay rate being 1% (or 0.01), and time being 10 weeks. The formula then becomes W = 244*(1 - 0.01)^10.
Explanation:Cal O'Ree's weight loss over the span of 10 weeks can be calculated using the principles of exponential decay. In this context, The doctor's statement implies that Cal's weight decreases by 1% each week - this is the decay rate. The solution to this kind of problem lies in the formula for exponential decay: W = P*(1 - r)^t, where P is the initial quantity (in this case, weight), r is the decay rate, and t is the time. For Cal, P = 244 pounds, r = 0.01, and t = 10 weeks.
After substituting these values into the formula, we get: W = 244*(1 - 0.01)^10. Calculating the expression gives us the weight that Cal is expected to have after 10 weeks of working out following the doctor's prognosis.
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n insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims. Calculate the probability that a claim submitted to the insurance company includes operating room charges.
Answer:
[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]
[tex]0.1 = 0.25 P(C)[/tex]
[tex] P(C) = 0.4[/tex]
Step-by-step explanation:
First we can define some notation useful:
C ="represent the event of incurring in operating charges"
R= represent the event of emergency rooms charges"
For this case we are interested on P(C) since they want "the probability that a claim submitted to the insurance company includes operating room charges."
We have some probabilities given:
[tex] P(R') = 0.25 , P(C \cup R) =0.85[/tex]
Solution to the problem
By the complement rule we have this:
[tex] P(R') = 0.25 =1-P(R)[/tex]
[tex] P(R) = 1-0.25 = 0.75[/tex]
Since the two events C and R are considered independent we have this:
[tex]P(C \cap R) = P(C) *P(R)[/tex]
Now we can use the total probability rule like this:
[tex] P(C \cup R) = P(C) + P(R) - P(R)*P(C)[/tex]
And if we replace we got:
[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]
[tex]0.1 = 0.25 P(C)[/tex]
[tex] P(C) = 0.4[/tex]
A hypothesis is _____ Question 20 options: a proven scientific fact an instrument that is used to examine environmental conditions a testable proposition that explains an observed phenomenon or answers a question the design of an experiment that can be used in scientific inquiry a prediction about something that has not yet been observed
The options are properly listed below:
A. a proven scientific fact
B. an instrument that is used to examine environmental conditions
C. a testable proposition that explains an observed phenomenon or answers a question
D. the design of an experiment that can be used in scientific inquiry
E. a prediction about something that has not yet been observed
ANSWER:
C. a testable proposition that explains an observed phenomenon or answers a question
EXPLANATION:
HYPOTHESIS is a proposed explanation for an observable fact, but that cannot satisfactorily be explained with the available scientific theories. It is based on the information acquired from a primary source or data collected during a scientific activity.
Also, for a hypothesis to be a scientific hypothesis, the scientific method requires that it is testable.
It is also a trial solution to a question.
Hypothesis can be provisionally accepted as a starting point for further research.
Suppose that there are 2 million inhabitants of a country in which 1 the mean gross (before tax) income is $20,000 per year. Suppose that the mean individual pays taxes on $3000 per year. What is the total disposable income (gross income minus taxes) per year for the country as a whole?
Answer:
$34 billion
Step-by-step explanation:
Data provided in the question:
Total number of inhabitants of a country = 2 million
1 mean gross income = $20,000 per year
Tax paid by mean individual = $3000 per year
Now,
Mean disposable income
= Mean gross income - Mean tax
= $20,000 - $3,000
= $17,000 per year
Therefore,
Total disposable income per year
= Mean disposable income × Total number of inhabitants of a country
= $17,000 × 2 million
= $34 billion
Select the most likely answer for the coefficient of linear correlation for the following two variables: x = the number of hours spent studying for a test, and y = the number of points earned on the test
a. r = 1.20b. r = 0.70c. r = - 0.85d. r = 0.05
Answer:
Option b) r = 0.70
Step-by-step explanation:
We are given the following in he question:
Variables:
x = the number of hours spent studying for a test
y = the number of points earned on the test
Correlation is a technique that help us to find or define a relationship between two variables. A positive correlation means that an increase in one quantity leads to an increase in another quantityA negative correlation means with increase in one quantity the other quantity decreases.+1 tells about a a perfect positive linear relationship and −1 indicates a perfect negative linear relationship.Values between 0 and 0.3 tells about a weak positive linear relationship, values between 0.3 and 0.7 shows a moderate positive correlation and a correlation of 0.7 and 1.0 states a strong positive linear relationship.Values between 0 and -0.3 tells about a weak negative linear relationship, values between -0.3 and -0.7 shows a moderate negative correlation and a correlation value of of -0.7 and -1.0 states a strong negative linear relationship.As the number of hours increases, the number of points earned on the test increases. Thus, the two variables are positively correlated.
Thus, the coefficient correlation between two variables can be given by r = 0.70, that shows a moderate positive correlation.
Option b) r = 0.70
Find the height of the ramp and the length of the base of the ramp
Answer:
x = 6.1 and y = 12.6
Step-by-step explanation:
sin 26 = x/14
x = 14sin26
x = 6.14
x= 6.1
cos 26 = y/14
y = 14cos26
y = 12.58
y = 12.6
Answer:the height of the ramp is 6.1ft
the length of the base of the ramp is 12.6 ft
Step-by-step explanation:
Triangle ABC is a right angle triangle.
The length of the ramp represents the hypotenuse of the right angle triangle.
With 26 degrees being the reference angle,
The opposite side of the right angle triangle = x
The adjacent side of the right angle triangle = y
To determine x, we would apply the Sine trigonometric ratio.
Sin θ = opposite side/hypotenuse
Sin 26 = x/14
x = 14Sin26 = 14 × 0.4384 =
x = 6.1
To determine y, we would apply the Cosine trigonometric ratio.
Cos θ = adjacent side /hypotenuse
Cos 26 = y/14
x = 14Cos26 = 14 × 0.8988 =
x = 12.6
Ace Truck leases its 10-ft box truck at $30/day and $0.50/mi, whereas Acme Truck leases a similar truck at $25/day and $0.55/mi.
(a) Find the daily cost of leasing from each company as a function of the number of miles driven.
Answer:
Ace Truck
[tex]C(m) = 30 + 0.5*m[/tex]
Acme Truck
[tex]C(m) = 25 + 0.55*m[/tex]
Step-by-step explanation:
The cost function to lease a box truck from a company has the following format:
[tex]C(m) = F + a*m[/tex]
In which F is the fixed cost and a is the cost per mile m.
(a) Find the daily cost of leasing from each company as a function of the number of miles driven.
Ace Truck
$30/day and $0.50/mi. This means that [tex]F = 30, a = 0.50[/tex]. So
[tex]C(m) = 30 + 0.5*m[/tex]
Acme Truck
$25/day and $0.55/mi. This means that [tex]F = 25 a = 0.55[/tex]. So
[tex]C(m) = 25 + 0.55*m[/tex]
drag each expression to show whether it can be used to find the surface area, volume or neither
The expressions can be used to calculate volume if they incorporate height, length, and width. They could calculate surface area if they involve length and width or the sum of all faces of a 3D object, without height. If these measurements are not present, they may not calculate surface area or volume.
Explanation:The original question appears to be a task related to a mathematical exercise instead of a traditional query. It's asking whether certain expressions can be used to calculate volume or surface area, or if they're not meant to calculate either.
If the expression used includes measurements for length, width, and height and involves their multiplication, it's likely being used to calculate the volume of a 3D object. For a cube, volume is calculated as length x width x height, for example.
If the expression incorporates the multiplication of length and width, without incorporating height, it's being used to calculate surface area. The expression might also calculate the sum of all the faces of a 3D object to find total surface area.
If there aren't any apparent calculations for length, width, or height, it could be that these expressions aren't being used for calculating surface area or volume.
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Students in a mathematics class were given an exam and then tested monthly with an equivalent exam. The average scores for the class are given by the human memory model f(t) = 76 − 18 log10(t + 1), 0 ≤ t ≤ 12 where t is the time in months. Verify your answers in parts (a), (b), and (c) using a graphing utility.
(a) What was the average score on the original exam (t = 0)? f(0) =
(b) What was the average score after 2 months? (Round your answer to one decimal place.) f(2) =
(c) What was the average score after 11 months? (Round your answer to one decimal place.) f(11) =
Answer:
a) [tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]
b) [tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]
c) [tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]
Step-by-step explanation:
For this case we know that the average scores for the class are given by the following model:
[tex] f(t) = 76-18 log_{10} (t+1) , 0 \leq t \leq 12[/tex]
Where t is in months. The graph attached illustrate the function for this case
And for this case we can answer the questions like this:
Part a
We just need to replace t =0 into the model and we got:
[tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]
Part b
We just need to replace t =2 into the model and we got:
[tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]
Part c
We just need to replace t =11 into the model and we got:
[tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]