Choose an American household at random, and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 6 cars:

Number of cars X Probability 0 1 2 3 4 5 6

0.07 0.31 0.43 0.12 0.04 0.02 0.01

A housing company builds houses with two-car garages. What percent of households have more cars than the garage can hold?

The expression 10∠30 + 10∠30 can be simplified by adding the magnitudes (10 + 10) and keeping the angle the same.

Given, that 10∠ 30 + 10∠ 30 .

In polar form, the magnitude is represented by the absolute value of a complex number and the angle is measured counterclockwise from the positive real axis.

To find the polar form of the sum, we first add the magnitudes: 10 + 10 = 20.

Next, keep the angle the same: 30 degrees.

Therefore, the polar form of 10∠30 + 10∠30 is 20∠30.

This means that the complex number is represented by a magnitude of 20 and an angle of 30 degrees.

Know more about polar form,

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Answer:

a) H0: mean =5 and Ha: mean≠ 5

Step-by-step explanation:

In hypothesis testing procedure the trait of null hypothesis is that it always contain an equality sign. We are known that diameter of spindle is known to be 5mm. This our null value. Hence the null hypothesis is

H0:μ=5.

Now for alternative hypothesis we are given that the mean diameter has moved away from the target. This means that mean diameter could be increases or decreases from 5mm. Hence the alternative hypothesis is

Ha:μ≠5

From the information given, it is found that the correct option is:

(A) H0: Mean= 5 and Ha: Mean is not equal to 5.

At the null hypothesis, it is tested if the motor works properly, that is, the spindle has diameter significantly close to 5 mm, hence:

[tex]H_0: \mu = 5[/tex]

At the alternative hypothesis, it is tested if the motor does not work properly, that is, the spindle has diameter different from 5 mm, either too high or too low, hence:

[tex]H_a: \mu \neq 5[/tex]

Thus, a is the correct option.

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Answer: The date in which the water level will not exceed 18ft is August 12 ( 6 weeks after July 1)

Step-by-step explanation:

Given:

Initial water level = 22ft

Final water level = 18ft

Total water level change = 22-18 = 4ft

Rate of change of water level = 2/3 ft/week

Using the formula;

Total change = rate × time

Time = total change/rate

Substituting the values of the rate and total water level change.

Time = 4/(2/3) week = 4 × 3 ÷2 = 6weeks.

From July 1 + 6 weeks = August 12

The date in which the water level will not exceed 18ft is August 12 ( 6 weeks after July 1)

Final answer:

To find the dates when the water level will not exceed 18 ft, we set up an inequality based on the given situation and solve it to determine the range of weeks when the water level stays below 18 ft.

Explanation:

To find the dates when the water level will not exceed 18 ft, we need to set up an inequality based on the given situation.

By solving the inequality, we can determine the range of weeks when the water level will not exceed 18 ft.

Answer: ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )

Step-by-step explanation:

Given : A random sample of 11 days were selected from last year's records maintained by the maternity ward in a local hospital, and the number of babies born each day of the days is given below:

3  7   7   10    0    7    1    2    5   3    0

We first arrange them in increasing order , we get

0    0    1    2   3   3   5     7   7   7   10

Here , N= 11

Now , we can see that

Minimum value = 0

Maximum value = 10

First quartile [tex]Q_1[/tex]= [tex](\dfrac{N+1}{4})^{th}\ term=(\dfrac{12}{4})^{th}\ term = 3^{rd} term =1[/tex]

Second quartile [tex]Q_2[/tex]= Median = Middlemost number = 3

Third quartile [tex]Q_3[/tex] =  [tex](\dfrac{3(N+1)}{4})^{th}\ term=(\dfrac{36}{4})^{th}\ term[/tex]

[tex]= 9^{th} term =7[/tex]

∴ The required five number summary : ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )

Answer: E. none of the above

Step-by-step explanation:

The given data values that represents the population:

2, 6, 3, and 1.

Number of values : n=4

Mean of the data values = [tex]\dfrac{\text{Sum of values}}{\text{No. of values}}[/tex]

[tex]\dfrac{2+6+3+1}{4}=\dfrac{12}{4}=3[/tex]

Sum of the squares of the difference between each values and the mean =

[tex](2-3)^2+(6-3)^2+(3-3)^2+(1-3)^2[/tex]

[tex]=-1^2+3^2+0^2+(-2)^2[/tex]

[tex]=1+9+0+4=14[/tex]

Now , Variance = (Sum of the squares of the difference between each values and the mean ) ÷ (n)

= (14) ÷ (4)= 3.5

Hence, the  variance is 3.5.  

Therefore , the correct  answer is "E. none of the above".

Answer:

x=-224/229,

y=296/229,

z=118/229

Step-by-step explanation:

-5x-7y-4z=-6

6x+2y+3z=-2

-x+2y-7z=0...........( multiple with (-5) and sum with 1st equation, mult with 6 and                    sum with 2nd equation)

______________

-x+2y-7z=0

-17y+31z=-6.....(mult with 14)

14y-39z=-2.....(mult with -17) then sum

___________

-x+2y-7z=0

-229z=-118, so here we have z=118/229.

14y-39*(118/229)=-2, from here we have y=296/229

-x+2*(296/229)-7*(118/229)=0, we get that x=-234/229

In the same way you can do this in the matrix form>>

To design a sine function with a period of 24 and minima and maxima at given times, we scale the function using a coefficient of 2π/24, shift the function to make the peak occur at x=16, and stretch it by a factor of 3 to make it go from 10 to 16.

To design a sine function with a period of 24 and minima and maxima at given times, we first need to understand a few concepts about sine functions. The standard sine function, sine(x), has a period of 2π. Therefore, to stretch it to a period of 24 hours, we would scale the function using a coefficient of 2π/24 or π/12. Thus, our function becomes sine((π/12) x).

Next, we want to shift the function so its maximum occurs at t=16. Normally, the sine function peaks at π/2, so we need to shift the function to the right by an amount that makes the peak occur at x=16. This would be 16 - π/2, which gives us the function sine(π/12 x - 16 + π/2).

Finally, to stretch the function vertically to accommodate the minimum and maximum values of 10 and 16, we note that the amplitude of the sine function is usually 1 (from -1 to 1), so we need to stretch it by a factor of (16-10)/2 = 3 to make it go from 10 to 16. This gives us the function y= 3sin((π/12)x - 16 + π/2)+13.

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Answer:

c. Observed data points that are far from the other observed data points in the horizontal direction

True, by definition are observed data points that are far from the other observed data points in the horizontal direction.

Step-by-step explanation:

When we conduct a regression we consider influential points by definition "an outlier that greatly affects the slope of the regression line". Based on this case we can analyze one by one the possible options:

a. Observed data points that are close to the other observed data points in the horizontal direction

False. If are close to the other observed values then are not influential points

b. Observed data points that are far from the least squares line

False, that's the definition of outlier.

c. Observed data points that are far from the other observed data points in the horizontal direction

True, by definition are observed data points that are far from the other observed data points in the horizontal direction.

d. Observed data points that are close to the least squares line

False. If are close to the fit regression line adjusted then not affect the general equation for the model and can't be considered as influential points

a.This set of vectors are not basis for vector space for two-dimentional space R2 due to high number of vectors (3). It means three vector is two much to span 2-dimentional space.

b.This set of vectors are not basis for vector space for three-dimentional space R3 due to small number of vectors (2). It means two vector can't span three-dimentional space.

Answer:

Stratified

Step-by-step explanation:

Stratified sampling is a type of sampling method in which the observer divides the overall population into different groups.

These groups are known as strata.

Within the each group, a probability sample is selected.

stratified sampling helps in reducing the sample size required to attain a required precision.

Final answer:

The probability of correctly choosing five random numbers from balls numbered 1 to 36 is calculated by multiplying the probability of choosing each number correctly, which is (1/36)^5, rounded to 0.0000 to four decimal places.

Explanation:

The student is asking about the probability of choosing five numbers that match five randomly selected balls when the balls are numbered 1 through 36. This is a question of combinatorial probability, where we are interested in the probability of one specific outcome in a set of possibilities.

To solve this, we need to calculate the probability of choosing each ball correctly. The probability of choosing the first number correctly is 1/36, since there is only one correct number out of 36. Likewise, the probability of choosing the second number correctly is also 1/36, and the same logic applies for the third, fourth, and fifth numbers. As these events are all independent (choosing one number does not affect the others), we can find the total probability by multiplying the individual probabilities together:

P(choosing all five numbers correctly) = P(choosing 1st number correctly) × P(choosing 2nd number correctly) × ... × P(choosing 5th number correctly) = (1/36)^5.

The exact value of this probability is quite small, and one would usually leave it as a fraction to avoid rounding errors. However, the instructions specify to round to four decimal places, so let's calculate:

(1/36)^5 = 1/60466176, which is a very small likelihood and as a decimal, it's approximately 0.0000000165, but you can rounded to 0.0000 when expressing it to four decimal places as per instruction.

Answer:

The numerical value is a parameter.                  

Step-by-step explanation:

We are given the following situation in the question:

A poll of all 2000 students in a high school found that Modifying 94 % with underline of its students owned cell phones.

Individual of interest:

Students in a high school found

Variable of interest:

Percentage of students owned a cell phone

Population of interest:

All 2000 high school students.

94 % with underline of its students owned cell phones.

Since this numerical value describes all of the 2000 high school student, it is describing the population of interest. Thus, the numerical value is a parameter.

Answer:

[tex] Range = 149-131=18[/tex]

[tex] s^2 =\frac{(131-139.5)^2 +(137-139.5)^2 +(138-139.5)^2 +(141-139.5)^2 +(141-139.5)^2 +(149-139.5)^2}{6-1}=35.1[/tex][tex] s =\sqrt{35.1}=5.9[/tex]

B. ​Ideally, the standard deviation would be zero because all the measurements should be the same.

Step-by-step explanation:

For this case we have the following data:

131 137 138 141 141 149

For this case the range is defined as [tex] Range = Max-Min[/tex]

And for our case we have [tex] Range = 149-131=18[/tex]

First we need to calculate the average given by this formula:

[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}=\frac{837}{6}=139.5[/tex]

We can calculate the sample variance with the following formula:

[tex] s^2 = \frac{\sum_{i=1}^n (X_i -\bar x)^2}{n-1}[/tex]

And if we replace we got:

[tex] s^2 =\frac{(131-139.5)^2 +(137-139.5)^2 +(138-139.5)^2 +(141-139.5)^2 +(141-139.5)^2 +(149-139.5)^2}{6-1}=35.1[/tex]

And the standard deviation is just the square root of the variance so then we got:

[tex] s =\sqrt{35.1}=5.9[/tex]

If the​ subject's blood pressure remains constant and the medical students correctly apply the same measurement​ technique, what should be the value of the standard​ deviation?

For this case the variance and deviation should be 0 since we not evidence change then we not have variation. And for this case the best answer is:

B. ​Ideally, the standard deviation would be zero because all the measurements should be the same.

Answer:

a) Quantitative

b) Quantitative

c) Qualitative

d) Qualitative

Step-by-step explanation:

a)

Percent of freshmen that will eventually graduate is a quantitative variable because it can be presented numerically for example 84% or 78% etc.

b)

GPA of incoming freshman is a quantitative variable because it can be presented by numerical quantities.

c)

Require SAT or ACT for admission is a categorical variable because it is divided into categories such as required, recommended and not used.

d)

College type is a categorical variable because it is divided into categories such as liberal arts college and national university etc.

The variables 'percent of freshmen who eventually graduate' and 'G.P.A of incoming freshmen' are quantitative because they can be measured numerically. The variables 'Require SAT or ACT for admission' and 'College type' are categorical because they are classified into specific categories.

In regards to the variables that contribute to the ranking of colleges, (a) The 'percent of freshmen who eventually graduate' can be identified as a QUANTITATIVE variable since it is a number that can be measured. For instance, an outcome could be '85% of freshmen graduate'.

(b) The 'GPAs of incoming freshmen' is also a QUANTITATIVE variable as it can also be measured numerically. An example could be 'The average GPA of incoming freshmen is 3.7 out of 4.0'.

(c) Whether or not a college 'requires SAT or ACT for admission' is a CATEGORICAL variable as it describes a category or characteristic, for example, the admission requirement can be one of the following: required, recommended or not used.

(d) Lastly, 'college type (liberal arts college, national university, etc.)' is also a CATEGORICAL variable, because it represents different types of educational institutions which are distinguished by categories.

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Answer:

a. mixed longitudinal study

True, by definition a mixed-longitudinal study is when "have defined some cohorts and these are followed for a shorter period and we can compare the precision, bias due to age and cohort effects" on the entire study. So that represent the perfect mix between longitudinal and cross sectional study.

Step-by-step explanation:

a. mixed longitudinal study

True, by definition a mixed-longitudinal study is when "have defined some cohorts and these are followed for a shorter period and we can compare the precision, bias due to age and cohort effects" on the entire study. So that represent the perfect mix between longitudinal and cross sectional study.

b. limited longitudinal study

This definition is not appropiate and is not usually used in the experimental designs.

c. longitudinal study

False, a longitudinal study is a design on which have "repeated observations of the same variables over short or long periods of time" and for this case we need cross sectional conditions, so for this case not applies.

d. cohort study

False, by definition a cohort study is an extesion of the longitudinal design but on this case " the samples are obtained from a cohort using cross-section intervals through time" and for this reason not applies for our case since we need a longitudinal design combined with the cross sectional design

Answer:

[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]

Step-by-step explanation:

We have been an integral [tex]\int \frac{9+\sqrt{x}+x}{x}dx[/tex]. We are asked to find the general solution for the given indefinite integral.

We can rewrite our given integral as:

[tex]\int \frac{9}{x}+\frac{\sqrt{x}}{x}+\frac{x}{x}dx[/tex]

[tex]\int \frac{9}{x}+\frac{1}{\sqrt{x}}+1dx[/tex]

Now, we will apply the sum rule of integrals as:

[tex]\int \frac{9}{x}dx+\int \frac{1}{\sqrt{x}}dx+\int 1dx[/tex]

[tex]9\int \frac{1}{x}dx+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]

Using common integral [tex]\int \frac{1}{x}dx=\text{ln}|x|[/tex], we will get:

[tex]9\text{ln}|x|+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]

Now, we will use power rule of integrals as:

[tex]9\text{ln}|x|+\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\int 1dx[/tex]

[tex]9\text{ln}|x|+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+\int 1dx[/tex]

[tex]9\text{ln}|x|+2x^{\frac{1}{2}}+\int 1dx[/tex]

[tex]9\text{ln}|x|+2\sqrt{x}+\int 1dx[/tex]

We know that integral of a constant is equal to constant times x, so integral of 1 would be x.

[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]

Therefore, our required integral would be [tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex].

The general indefinite integral of the expression 9 + sqrt(x) + x/x is obtained by separately integrating each term, resulting in the expression 9x + (2/3)x^(3/2) + x + C, where C is the sum of the constants of integration.

The question asks to compute the indefinite integral of an algebraic function. Firstly, it's crucial to understand that an indefinite integral is an antiderivative of a function, representing the reverse operation of differentiation. In the given expression, we essentially have three terms to integrate: 9, √x and x/x (which simplifies to 1). To find the integral of these expressions we can use the power rule of integration: ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration.

When we integrate 9, treating it as 9x^0, it becomes 9x + C1. The integral of the square root of x, which is x^1/2 in exponent form, becomes (2/3)x^(3/2) + C2 according to the power rule. Finally, x/x simplifies to 1, and its integral is simply x + C3.

So, the general indefinite integral of the original expression is 9x + (2/3)x^(3/2) + x + C, where 'C' represents the sum of the constants of integration: C1, C2, and C3.

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Answer:

See explanation below.

Step-by-step explanation:

Part a

For this case we can use the moment generating function for the bernoulli distribution for n trials, given by:

[tex] p(s)^n = (q+ps)^n =\sum_{r=0}^n (nCr) (ps)^r q^{n-r}[/tex]

Wehre p is the probability of success and [tex] P(s_ = q+ps[/tex]

Using this property we see that;

If we multiply the two generating functions we got:

[tex] p(s)^N = (q+ps)^N =\sum_{r=0}^N (NCr) (ps)^r q^{N-r}[/tex]

[tex] p(s)^M = (q+ps)^M =\sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]

[tex] p(s)^M p(s)^N = \sum_{r=0}^N (NCr) (ps)^r q^{N-r} \sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]

And the mass function would be given by:

[tex]= (N+M C r) p^{r} (1-p)^{N+M-r}[/tex]

So we see that follows a binomial random variable with parameters (N+M, p)

Part b

For this case we are assuming that [tex] X \sim Bin (N,p) , Y\sim Bin (M,p)[/tex] and for this case we can assume that [tex] 0 \leq k \leq N+M[/tex] for the proof.

We are interested on the random variable [tex] Z= X+Y[/tex] since the two random variables are independent we can write the probability mass function for Z like this:

[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i , Y=k-i)[/tex]

[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i) P(Y=k-i)[/tex]

And we can replace the mass function for X and Y

[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} \sum_{i=0}^k (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]

And we can rewrite this like that:

[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]

[We can take out the constant p:

[tex] P(Z = X+Y = k) = p^k (1-p)^{N+M-k} \sum_{i=0}^k (NCi)(M C k-i)[/tex]

And using properties of the binomial formula we can write this like that:

[tex] P(Z = X+Y = k) = (N+M Ck) p^k (1-p)^{N+M-k} [/tex]

So then we see that [tex] Z= X+Y \sim Bin(N+M ,p)[/tex]

Answer:

Option C: The line y = x

Step-by-step explanation:

Let us take an ordered pair (x, y) of a function. Then the ordered pair of its inverse function would be (y, x).

That is to say, when we reflect a point (x, y) across the line y = x we get the point (y, x).

Note that since, this function is invertible, it is both 'one - one' and 'onto'.

The line of reflection for two functions with an inverse relationship is the line y = x, as this line shows the symmetry of the original function and its inverse on a graph.

When two functions have an inverse relationship, the line of reflection across which their graphs are mirrored is the line y = x. This is because for an inverse relationship, each x value in the first function corresponds to a y value in the second function, and vice versa. Thus, when graphing the original function and its inverse, you will notice that they are symmetric with respect to the line y = x.

Answer:

a) ∫_{-6}^{6} ∫_{0}^{36} ∫_{x²}^{36} (-y) dy dz dx

b)  ∫_{0}^{36} ∫_{-6}^{6} ∫_{x²}^{36} (-y) dy dx dz

c)  ∫_{0}^{36} ∫_{x²}^{36} ∫_{-6}^{6} (-y) dx dy dz

e) ∫_{x²}^{36} ∫_{-6}^{6} ∫_{0}^{36} (-y) dz dx dy

Step-by-step explanation:

We write the equivalent integrals for given integral,

we get:

a) ∫_{-6}^{6} ∫_{0}^{36} ∫_{x²}^{36} (-y) dy dz dx

b)  ∫_{0}^{36} ∫_{-6}^{6} ∫_{x²}^{36} (-y) dy dx dz

c)  ∫_{0}^{36} ∫_{x²}^{36} ∫_{-6}^{6} (-y) dx dy dz

e) ∫_{x²}^{36} ∫_{-6}^{6} ∫_{0}^{36} (-y) dz dx dy

We changed places of integration, and changed boundaries for certain integrals.

Answer:

Option D : 3 Superscript negative x Baseline = 3 Superscript 3 x + 6

Step-by-step explanation:

Let us first convert all the equations in Mathematical form for readability.

The question equation will become:

[tex](\frac{1}{3})^{x}=(27)^{x+2}[/tex]        ----------------- (1)

And the option equations will be:

A.   [tex]3^{x}=3^{(-3x+2)}[/tex]

B.   [tex]3^{x}=3^{(3x+6)}[/tex]

C.   [tex]3^{-x}=3^{(3x+2)}[/tex]

D.   [tex]3^{-x}=3^{(3x+6)}[/tex]

Now, let's solve the question equation. Simplifying equation (1), we get

[tex](3^{-1})^{x} = (3^3)^{x+2}\\\\3^{-x} = 3^{3(x+2)}\\\\3^{-x} = 3^{(3x+6)}[/tex]

Hence, option D is correct.

Answer:

D

Step-by-step explanation:

Answer:

Option D) There is sufficient evidence that the true population proportion is not equal to 70%.        

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 1165

p = 70% = 0.7

Alpha, α = 0.02

Number of people who agreed , x = 746

First, we design the null and the alternate hypothesis  

[tex]H_{0}: p = 0.7\\H_A: p \neq 0.7[/tex]

This is a two-tailed test.  

Formula:

[tex]\hat{p} = \dfrac{x}{n} = \dfrac{746}{1165} = 0.64[/tex]

[tex]z = \dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]

Putting the values, we get,

[tex]z = \displaystyle\frac{0.64-0.7}{\sqrt{\frac{0.7(1-0.7)}{1165}}} = -4.46[/tex]

Now, [tex]z_{critical} \text{ at 0.02 level of significance } = \pm 2.33[/tex]

Since,  

Since, the calculated z statistic does not lie in the acceptance region, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.

Thus, there is enough evidence to support the claim that the true proportion differs from 70%.

Option D) There is sufficient evidence that the true population proportion is not equal to 70%.

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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Answer: Ф = 20

Step-by-step explanation:

sin2Ф = cos (Ф+ 30 )

if sinα = cosβ , then α and β are complementary ,that is they add up to be [tex]90^{0}[/tex].

Therefore : 2Ф + Ф + 30 = 90

3Ф + 30 = 90

3Ф         = 90 - 30

3Ф        = 60

Ф          = 20

Answer:

1. First equation is option B

2. Second equation is option C

3. Third equation is option E

4. Fourth equation no best option.

explanation:

Check the attachment for solution

Following are the calculation to the differential equation:

For point 1)

[tex]y'' + 8 y' + 15 y = 0\\[/tex]

B

[tex]Y = e^{-3x}[/tex] be the solution of this equation

[tex]Y' = -3 e^{-3x}\\\\ y''= 9 e^{-3x} \\\\\therefore \\\\y'' +8y' + 15 y= 9e^{-3x} + 8(-3e^{-3x})+ 15 e^{-3x} \\\\e^{-3x}( 9-24+15)=0[/tex]

For point 2)  

[tex]y'' + y = 0[/tex]

C

[tex]y = \sin x[/tex] be the solution of above equation  

[tex]y'= -\cos x \\\\y''= -\sin x = -y \\\\y''+y=0\\\\[/tex]

For point 3)

 [tex]y' = 5 y[/tex]

[tex]y'=e^{5x}[/tex] be the solution of equation 3

[tex]y'= 5 e^{5y}= 5y =y'=5y[/tex]

For point 4)

[tex]2x^2 y'' + 3xy' = y[/tex]

Let [tex]y=\sqrt{x}[/tex] be the solution of equation  (4)

 [tex]y'=\frac{1}{2 \sqrt{x} }\\\\y''=- \frac{1}{2} \times \frac{1}{2} \times {x^{- \frac{3}{2}}} ==- \frac{1}{4} \times {x^{- \frac{3}{2}}} \\\\-2x^2 \times =- \frac{1}{4} {x^{- \frac{3}{2}}}+ 3x \times =- \frac{1}{2 \sqrt{x}}\\\\- \frac{1}{2} {x^{ \frac{1}{2}}}+ \frac{3}{2} x^{\frac{1}{2}} =\sqrt{x} =y\\\\[/tex]

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The value of  A and B given that the function [tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex] has a minimum value of 54 at x = 81 is 243 and 3 respectively

Given the function

[tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex]

If y= 54 where x = 81, hence

[tex]54=\frac{A}{\sqrt{81} }+B\sqrt{81}\\54=\frac{A}{9}+9B\\486=A+81B\\ A+81B=486[/tex]

At the minimum point [tex]\frac{dy}{dx} = 0[/tex]

Differentiate the given function:

[tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}\\y'=\frac{-0.5A}{{x^{3/2}} }+\frac{B}{x^{1/2}} \\\frac{-0.5A}{{x^{3/2}} }+\frac{B}{x^{1/2}}=0[/tex]

Substitute x = 81 to hav:

[tex]\frac{-0.5A}{81^{2/3}} +\frac{B}{81^{1/2}}=0\\\frac{-A}{81} + B=0\\-A+81B=0\\A=81B ......................... 2[/tex]

Substitute equation 2 into 1:

[tex]81B+81B= 486\\162B=486\\B=\frac{486}{162} \\B=3[/tex]

Get the value of A:

[tex]A=81B\\A=81(3)\\A=243[/tex]

Hence the value of  A and B given that the function [tex]y=\frac{A}{\sqrt{x} }+B\sqrt{x}[/tex] has a minimum value of 54 at x = 81 is 243 and 3 respectively

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The correct values for A and B, given that the function y=Ax√+Bx√ has a minimum value of 54 at x = 81, would be A=243 and B=6. The sum of both divided by 81 equals 54, as stated in the equation.

The function provided in this Mathematics problem is y = Ax√ + Bx√:

We are told that function has a minimum value of 54 at x = 81. So if we insert 81 into x, we would get:

54 = 81A + 81B

Then simplify:

54 = 81(A + B)

To find the value for A and B, we need to know more about the relationship between A and B. Unfortunately, the problem doesn't supply enough information for us to determine exact figures of A and B. But from the options provided, we need A and B that sum up to 54/81. Of the choices provided, only (A = 243, B = 6) will give us that sum, so b.) is the correct choice.

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Answer:

[tex] Q_1 = 24[/tex]

[tex]Q_3 = 29[/tex]

[tex] IQR= Q_3 -Q_1 = 29-24 =5[/tex]

Step-by-step explanation:

For this case we have the following dataset:

25,21,26,24,29,33,29,25,19,24

The first step is order the data on increasing order and we got:

19, 21, 24, 24, 25, 25, 26, 29, 29 , 33

For this case we have n=10 an even number of data values.

We can find the median on this case is the average between the 5 and 6 position from the data ordered:

[tex] Median = \frac{25+25}{2}=25[/tex]

In order to find the first quartile we know that the lower half of the data is: {19, 21, 24, 24, 25}, and if we find the middle point for this interval we got 24 so this value would be the first quartile [tex] Q_1 = 24[/tex]

For the upper half of the data we have {25,26,29,29,33} and the middle value for this case is 29 and that represent the third quartile [tex]Q_3 = 29[/tex]

And finally since we have the quartiles we can find the interquartile rang with the following formula:

[tex] IQR= Q_3 -Q_1 = 29-24 =5[/tex]

Answer:

im pretty sure you have to find out how far Joel walked and then double joels distance to get Brents

Step-by-step explanation:

Answer:

Step-by-step explanation:

Final answer:

The described observational study is a cross-sectional study, as it involves collecting data once at a particular point in time, without any follow-up observations. So the correct option is B.

Explanation:

The type of observational study described in the scenario is a cross-sectional study. This classification is based on the researchers collecting data at a specific point in time by interviewing a sample of 1800 adults about their tendencies in handling emotions and measuring their blood pressure. Since the data on suppression of emotions and blood pressure is gathered just once from these individuals, without any follow-up at later dates, this defines the hallmark of a cross-sectional study. In contrast, a retrospective study would involve looking back at past behaviors or conditions, and a cohort study would imply following a group of individuals over a prolonged period of time to observe outcomes, neither of which applies to this particular research setup.

Final answer:

The study in the question is a cross-sectional study because it gathers data from subjects at a specific point in time to assess the association between emotional suppression and high blood pressure.

Explanation:

The study described in the question can be classified as a cross-sectional study because it involved collecting data from the individuals at a specific point in time. Participants in the study were asked about their response to emotions and their current tendency to express or hold in anger and other emotions. Additionally, their blood pressure was measured. This type of observational study is designed to analyze data at a single point in time to find any associations or correlations, such as the one being investigated between emotional suppression and high blood pressure.

Answer: Spatial correlation is defined as a specific relationship between spatial proximity among observational units and numeric similarities among values.

Step-by-step explanation: Spatial analysis focuses on individual as units located in spatial oriented structures such as gangs. In crime mapping and behavioral studies, spatial autocorrelation is used to determine the adjacency between area of influence and individuals within an area.

Answer:

2.15% was the percent error of the 2007 measurement.

Step-by-step explanation:

To calculate the percentage error, we use the equation:

[tex]\%\text{ error}=\frac{|\text{Experimental value - Theoretical value}|}{\text{Theoretical value}}\times 100[/tex]

We are given:

Experimental value of diameter of Pluto ,2015= 2372 km

Theoretical value of diameter of Pluto, 2007 = 2322 km

Putting values in above equation, we get:

[tex]\%\text{ error}=\frac{|2372 km-2322 km|}{2322 km}\times 100\\\\\%\text{ error}=2.15\%[/tex]

Hence, 2.15% was the percent error of the 2007 measurement.

Final answer:

The percent error of the 2007 measurement of Pluto's diameter is 2.11%.

Explanation:

The percent error can be calculated by using the formula:

Percent Error = [(Measured Value - Actual Value) / Actual Value] × 100%

Given that the measured diameter of Pluto in 2007 was 2322 km and the actual diameter is 2372 km, we can substitute these values into the formula to calculate the percent error.

Percent Error = [(2372 km - 2322 km) / 2372 km] × 100% = 2.11%