there was a substance of a 450 milligrams of radioactive substance to start a study. Since then the sample has decayed of 5.4% each year. Let t be the number of years since the start of the study. Let Y be the mass of the sample in milligrams. Write an exponential function showing the relationship of the y and t
Answer: [tex]\bold{Y = 450e^{(.054t)}}[/tex]
Step-by-step explanation:
The general formula for decay is:
[tex]A = Pe^{rt}[/tex] ; where:
P is the initial massr is the rate (in decimal form)t is the timeGiven:
P = 450r = 5.4% = .054Equation:
[tex]Y = 450e^{(.054t)}[/tex]
Translate the graph according to the rule (x, y) → (x – 2, y). The first graph goes with the question.
Look at the picture.
(x, y) → (x + 2, y)
Translate the graph of f(x) 2 units right.
-------------------------------------------------------------
(x, y) → (x, y + n) - translate the graph n units up
(x, y) → (x, y - n) - translate the graph n units down
(x, y) → (x - n, y) - translate the graph n units left
(x, y) → (x + n, y) - translate the graph n units right
PLEASE ANSWER THIS QUESTION !! 30 POINTS AND BRAINLIEST TO THE FIRST PERSON!
Answer:
x<=11
Step-by-step explanation:
2(x-3 ) <= 16
Distribute the 2
2x -6 <= 16
Add 6 to each side
2x -6+6<= 16+6
2x <= 22
Divide by 2
2x/2 <= 22/2
x<=11
The function f(x) goes through the point (2, 6). What point will this translate to in the function f(x) = (x + 2) – 3?
(0, 9)
(0, 3)
(4, 3)
(4, 9)
Answer:
(0,3)
Step-by-step explanation:
g(x) = f(x + 2) – 3
This is a shift of 2 to the left and a shift of 3 down
x moves from 2 to 2 units left (-2) so it becomes 0
y moves from 6 to 3 units down (-3) so it becomes 3
Answer:
the answer is B. or (0,3)
Step-by-step explanation:
The expression shown is the cost a customer pays for an item, where c is the cost the store pays for the item and 0.85c is the 85% price increase the store adds to the item
Answer:0.85c
Step-by-step explanation:
Answer:
0.85c
Step-by-step explanation:
i did the diagnostic
A contaminant is found in a solution at a level of 3/500%. What fraction of the solution is this
Answer:3/5
Step-by-step explanation:
First we change 500% to fractions
Changing percentage to fraction is dividing the number by 100
500% = 500/100
Putting this we get 3/(500/100)
Then when a number is being divided by a fraction, to get the answer, we multiply the number by the inverse of the fraction
3/(500/100) = 3 x 100/500 = 300/500 = 3/5
If contaminant is found in a solution at a level of 3/500%. Then 3/5 is fraction of the solution
What is Fraction?A fraction represents a part of a whole.
Given,
A contaminant is found in a solution at a level of 3/500%
Let us convert five hundred percentage to a fraction.
500% is converted to fraction by dividing 500/100
So let us divide three by 500/100
3/ (500/100)
When a fraction is divided with another fraction, then the denominator is multiplied inversely with numerator
3×100/500
=300/500
3/5
Hence if contaminant is found in a solution at a level of 3/500%. Then 3/5 is fraction of the solution
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Write a linear function for the line represented by the point-slope equation
y −4 = 3 (x − 4)
Must show your work to receive full credit.
Answer:
f(x) = 3x-8
Step-by-step explanation:
y −4 = 3 (x − 4)
We want to write the equation in y = mx+ b form
Distribute the 3
y-4 = 3x -3*4
y-4 = 3x -12
Add 4 to each side
y -4+4 = 3x-12+4
y = 3x-8
Since the want it in function form, f(x)
f(x) = 3x-8
Identify the value of p. Give your answers in simplest radical form. HELP PLEASE!!
Answer:
C
Step-by-step explanation:
using the cosine ratio to find p
cos 45° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{6}{p}[/tex]
cross- multiplying gives
p × cos45° = 6 → [ cos 45° = [tex]\frac{1}{\sqrt{2} }[/tex]]
p × [tex]\frac{1}{\sqrt{2} }[/tex] = 6
multiply both sides by [tex]\sqrt{2}[/tex]
⇒ p = 6[tex]\sqrt{2}[/tex] ( third option on list )
To determine the value of p in simplest radical form, solve the equation by simplifying the denominator and taking the square root of both sides. The value of p is 2.
Explanation:To identify the value of p, we need to solve the equation and simplify it to the simplest radical form. Let's start by simplifying the denominator and expressing it as a perfect square.
Next, we can take the square root of both sides of the equation to eliminate the square. This will help us isolate p. Once we do that, we can solve for p by getting rid of the square root on the left side of the equation. After simplifying, we find that p = 2.
So, the value of p is 2.
Tossing coins imagine tossing a fair coin 3 times. (a) what is the sample space for this chance process? (b) what is the assignment of probabilities to outcomes in this sample space?
Answer:
We are given that a fair coin is tossed 3 times.
We know that if a fair coin is tossed 3 times, then there are 8 possible outcomes.
(a) what is the sample space for this chance process?
The sample space associated with tossing a fair coin three times are:
Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Where:
H denotes the head and T denotes the tail.
(b) what is the assignment of probabilities to outcomes in this sample space?
We are given that a fair coin is tossed 3 times, which means that all the possible outcomes in the above mentioned sample space has equal chance being selected. Therefore, the assignment of probabilities to outcomes in this sample space is same for all outcomes and is given below:
[tex]p= \frac{1}{8}[/tex]
Officer Brimberry wrote 24 tickets for traffic violations last week, but only 21 tickets this week. What is the percent decrease? Give your answer to the nearest tenth of a percent.
Answer:3 percent because
Answer:
Percent = 62.5%
Step-by-step explanation:
A store is going out of business. Everything is marked down 40%. How much do you pay now for an item that used to cost $150?
A. What is the sum of the squares of the roots of $x^2 - 5x - 4 = 0$?
B. One root of $x^2 + 12x + k = 0$ is twice the other root. Find $k.$
C. What is the sum of the roots of the quadratic $4x^2 - 4x - 4$?
D. Jimmy is trying to factor the quadratic equation $ax^2 + bx + c = 0.$ He assumes that it will factor in the form
\[ax^2 + bx + c = (Ax + B)(Cx + D),\]where $A,$ $B,$ $C,$ and $D$ are integers. If $a = 4,$ and Jimmy wants to find the value of $A,$ what are the possible values he should check, in order to find $A$?
E. Brandy is trying to factor the quadratic $3x^2 - x - 10.$ She starts by assuming that the quadratic factors as
\[3x^2 - x - 10 = (x + B)(3x + D),\]for some integers $B$ and $D.$ After some work, Brandy successfully factors the quadratic. Find the ordered pair $(B,D).$
Answer:
A. 33
B. k=32
C. 1
D. [tex]\pm 1,\ \pm 2,\ \pm 4[/tex]
E. [tex]B=-2,\ D=5[/tex]
Step-by-step explanation:
In all parts for the quadratic equation [tex]ax^2+bx+c=0[/tex] use Vieta's formulas
[tex]x_1+x_2=-\dfrac{b}{a},\\ \\x_1\cdot x_2=\dfrac{c}{a},[/tex]
where [tex]x_1,\ x_2[/tex] are the roots of the quadratic equation.
A. For the equation [tex]x^2-5x-4=0,[/tex]
[tex]x_1+x_2=5,\\ \\x_1\cdot x_2=-4.[/tex]
Then
[tex](x_1+x_2)^2=x_1^2+2x_1\cdot x_2+x_2^2,\\ \\5^2=x_1^2+x_2^2+2\cdot (-4),\\ \\x_1^2+x_2^2=25+8=33.[/tex]
B. One of the roots of [tex]x^2+12x+k=0[/tex] is twice the other root, then [tex]x_2=2x_1.[/tex] By the Vieta's formulas,
[tex]x_1+x_2=3x_1=-12,\\ \\x_1\cdot x_2=2x_1^2=k.[/tex]
Then [tex]x_1=-4[/tex] and [tex]k=2x_1^2=2\cdot (-4)^2=2\cdot 16=32.[/tex]
C. The sum of the roots of the quadratic [tex]4x^2-4x-4[/tex] is [tex]-\dfrac{b}{c}=-\dfrac{-4}{4}=1.[/tex]
D. Note that
[tex](Ax+B)(Cx+D)=ACx^2+x(AD+BC)+BD,[/tex]
then [tex]AC=a=4.[/tex] If [tex]A,\ B,\ C,\ D[/tex] are integers, then you should check [tex]A=\pm 1,\ \pm 2,\ \pm 4.[/tex]
E. Consider [tex]3x^2 - x - 10 = (x + B)(3x + D).[/tex] Note that
[tex]x_1+x_2=\dfrac{1}{3},\\ \\x_1\cdot x_2=-\dfrac{10}{3}.[/tex]
Then
[tex]x_1=2,\ x_2=-\dfrac{5}{3}.[/tex]
Then [tex]3x^2 - x - 10 = (x -2)(3x+5),[/tex] hence [tex]B=-2,\ D=5.[/tex]
A. The sum of the squares of the roots is 33, B. k = 32, C. The sum of the roots is 1, D. The possible values he should check are ±1, ±2, ±4, E. The ordered pair (B, D) is (2, -5).
Let's solve each part of the problem step-by-step:
A. First, find the roots of the quadratic equation using Vieta's formulas:
The sum of the roots (α + β) = -(-5) = 5.The product of the roots (αβ) = -4.Now, the sum of the squares of the roots is given by: (α² + β²) = (α + β)² - 2αβ. Substituting the known values:
Hence, α² + β² = 25 - (-8) = 25 + 8 = 33.
B. Let the roots be α and 2α. Using Vieta's formulas again:
The sum of the roots (α + 2α) = 3α = -12, thus α = -4.The product of the roots (α * 2α) = 2α² = k.Therefore, k = 2(-4)² = 2 * 16 = 32.C. The sum of the roots is given by -b/a:
Here, a = 4 and b = -4.Thus, the sum of the roots = -(-4)/4 = 1.D. The quadratic can be written as 4x² + bx + c. The coefficient of x² on the right-hand side must be AC. Since a = 4:
The possible integer values for A can be the pairs (A, C) where A * C = 4.Thus, the possible values for A are ±1, ±2, ±4.E. The quadratic can be factored as (x + B)(3x + D). Let's determine B and D:
3B + D = -1 (coefficient of x)BD = -10 (constant term)Solving these equations, we find B = 2 and D = -5.Thus, the ordered pair (B, D) is (2, -5).
HELP ASAP PLZ!!! Collin deposited $5,500 in a savings account that earns 4.5% simple annual interest. The formula that can be used for calculating simple interest is I=prt , where I represents interest, p is the principle, r is the rate and t is the time. How much interest is earned after 5 years if he makes no other deposits or withdrawals?
Answer:
$1237.50
Step-by-step explanation:
r = 4.5% = 0.045
I = prt
I = 5500*0.045*5
= $1237.50
the consumer price index compares the cost of goods and services over various years, where 1967 is used as t=0. The same goods that cost $100 in 1967 cost $184.50 in 1977. Find an exponential function to model this data. Estimate what those goods would cost in 2005
Answer:
f(t) = 100·1.845^(t/10)$1025.15Step-by-step explanation:
(a) The given numbers can be put directly into the form ...
... f(t) = (initial value) · (ratio)^(t/(time to achieve that ratio))
Here, we have an intial value of $100, and a ratio of $184.50/$100 = 1.845. The time to achieve that multiplication is 10 years (1967 to 1977). So, the equation can be written ...
... f(t) = 100·1.845^(t/10)
(b) You want to find f(38).
... f(38) = 100·1.845^(38/10) = 100·1.845^3.8 ≈ 1025.15 . . . dollars
Final answer:
Using the provided CPI data, an exponential function is found to model the change in cost of goods over time. The estimated cost of the same goods in 2005, using this model, is approximately $1019.03, illustrating changes in purchasing power and the cost of living.
Explanation:
The question involves finding an exponential function to model the Consumer Price Index (CPI) data and estimate the cost of goods in a future year based on past data. The data provided includes the cost of the same goods being $100 in 1967 (t=0) and $184.50 in 1977. An exponential function of the form y = abt can be used, where y represents the cost of goods, t represents the year, and a and b are constants to be found.
Given that the goods cost $100 in 1967, our initial condition gives us a as $100. We then use the information from 1977 (t=10) to solve for b. Plugging in the values, we get $184.50 = 100*b10, which solves to b approximately equal to 1.0653. Therefore, the exponential function modeling the CPI data is y = 100(1.0653)t.
To estimate the cost of goods in 2005, we substitute t with 38 (2005 - 1967), resulting in an estimated cost of goods y ≈ 100(1.0653)³⁸ ≈ $1019.03. This estimation illustrates how the Consumer Price Index can indicate changes in purchasing power and the cost of living over time.
Combine and simplify these radicals. square root of 3 times the square root of 16
Answer:
[tex]\sqrt{3} * /sqrt{16} = 4\sqrt{3}[/tex]
Step-by-step explanation:
Answer:
Step-by-step explanation:
square root of 3 times the square root of 16 looks like this
[tex]\sqrt{3} .\sqrt{16}[/tex]
we try to break the numbers inside the radical to there prime factors
so
[tex]\sqrt{3} .\sqrt{2.2.2.2}[/tex]
the prime factors which are in pair , comes out of the radical so
[tex]\sqrt{3} .2.2[/tex]
which simplifies to
[tex]4\sqrt{3}[/tex]
The directions on a sewing pattern say to cut an extra 15% of fabric to account for error. Chloe needs 0.75 yard of frantic to make a skirt, and she cuts 0.1125 yard. Did Chloe cut the right amount of fabric?
Answer:
Yes she did
Step-by-step explanation:
There is .75 yards and she cut .1125 of it, it make it 15% as .75 x .15 = .1125
47=16384 write in logarithmic form
Answer:
[tex]\log_4{(16384)}=7[/tex]
Step-by-step explanation:
The base of the exponent is the base of the logarithm. The exponent is the logarithm. The value the exponential expression is equal to is the argument of the logarithm.
4 Questions about The Polynomial Remainder Theorem
None of these questions have anything to do directly with the polynomial remainder theorem. The theorem says that the remainder upon dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] is given by the value of [tex]f(c)[/tex].
For these questions, all you really have to do is evaluate the given polynomials at the given points, and IMO is much less work.
Question 2: [tex]f(5)=4(5)^2-5(5)+2=77[/tex]
Question 3: [tex]f(4)=3(4)^4-8)4)^2-2(4)+12=644[/tex]
Question 4: Here you have check the value of [tex]h(x)[/tex] and 2 and -2, then interpret them as points in the coordinate plane, [tex](x,h(x))[/tex].
[tex]h(-2)=2(-2)^5-4(-2)^4-2(-2)^2+15=-121[/tex]
[tex]h(2)=2(2)^5-4(2)^4-2(2)^2+15=7[/tex]
Question 5: Same as in question 4, but you have to check [tex]h(x)[/tex] at -4, -3, -2, -1.
[tex]h(-4)=72[/tex]
[tex]h(-3)=46[/tex]
[tex]h(-2)=26[/tex]
[tex]h(-1)=12[/tex]
- - -
If you insist on using the polynomial remainder theorem, it's a question of polynomial division. For instance, in question 2 you'd compute
[tex]\dfrac{4x^2-5x+2}{x-5}=4x+15+\dfrac{77}{x-5}\implies4x^2-5x+2=(4x+15)(x-5)+77[/tex]
so the remainder is 77, as we found by simply computing [tex]f(5)[/tex].
Kasia wants to create a triangle with angles measuring 37 degrees, 62 degrees, and 81 degrees. She says that only one unique triangle is possible given these angle measures. Which statement about Kasia’s claim is true?
Kasia is incorrect. Since the angles sum to 180 degrees and no side lengths are given, multiple triangles can be drawn with these angle measures.
Kasia is incorrect. No triangle can be made with these angle measures because they do not sum to 180 degrees.
Kasia is correct. Since the angles sum to 180 degrees and no measurements are given, only one unique triangle is possible.
Kasia is correct. Since the angles do not sum to 180 degrees, only one unique triangle is possible.
Answer:
option: A is correct
Step-by-step explanation:
kaisa's claim is incorrect. clearly the sum of the three angles is 180 degrees therefore we could easily create a triangle with the help of this information but we can't say that this traingle will be unique so we need to know some more information regarding its side.hence multiple triangles could be drawn with these angle measures.
hence option A is correct.
Answer:
A
Step-by-step explanation:
sofia bought bananas ,cereal,and milk at the store.She spent all of her money.She spent 3/10 of her money on bananas and 4/10 on cereal.What fraction of her money did Sofia spend on milk?Write and solve equations.
Please help me out! :) !!!!!!
Answer:
your answer is 8/40 but is reduced to 1/5
Step-by-step explanation:
A point of sale transaction occurs when an ATM withdrawal is made. Please select the best answer from the choices provided T F
Answer:
A point of sale transaction occurs when an ATM withdrawal is made- TRUE.
Step-by-step explanation:
A point of sale transaction occurs when an ATM withdrawal is made - This is TRUE.
A point of sale is the point when a transaction is finalized. This transaction is based on any form of payment like - cash, debit cards, credit cards etc.
Please help meeee!!!
Answer:
26, 37
Step-by-step explanation:
1,3,5,7,9,11
17 + 9 = 26
26 + 11 = 37
The price of milk tripled, and then rose another $0.75 per gallon. If the price now is AT LEAST $4.50 per gallon, which inequality expresses this situation? A) 3x + 0.75 ≤ 4.5 B) 3x + 0.75 ≥ 4.5 C) 3x - 0.75 ≥ 4.5 D) 3(x + 0.75) ≤ 4.5
This can be represented by the inequality:
3x + 0.75 ≥ 4.5
Inequality shows the non-equal comparison between two numbers or mathematical expressions.
Let x represent the price of milk per gallon.
Since the price of milk tripled, and then rose another $0.75 per gallon, hence:
Price of milk = 3x + 0.75
The price is now is AT LEAST $4.50 per gallon, hence this can be represented by the inequality:
3x + 0.75 ≥ 4.5
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Which number is written in scientific notation?
A) 7.8 x 10^-5
B) 3.4 x 100^2
C) 0.84 x 10^6
D) -5 x 10^-12
Answer:
I would say A. 7.8 x 10^-5
Step-by-step explanation:
7.8 x 10^-5 can also be written as 7.8 times 10 to the negative fifth power or the exponent of negative five. But, if you solve it you would get 0.000078 as the real number.
What is the slope of a line parallel to the line with equation 5x + 3y = 7?
Answer:
-5/3
Step-by-step explanation:
Parallel lines are lines which have the exact same slope but different y-intercepts. We can find the slope by converting to slope-intercept form, y=mx+b from standard form.
We convert by using inverse operations to isolate y.
5x+3y=7
5x-5x+3y=7-5x
0x+3y=-5x+7
3y=-5x+7
[tex]\frac{3y}{3} =\frac{-5x+7}{3} \\y=\frac{-5}{3}x+\frac{7}{3}[/tex].
The slope is -5/3. SInce parallel lines have the same slope, the slope for a parallel line will be -5/3.
Donna used 30 buttons of different colors and sizes to make a design. She used 12 large blue buttons the rest were small and yellow or small and green there were the same number of yellow and green buttons how many buttons were small and yellow
Fill in the value of the function, when the input is 2: y = −4 + 2
Answer:
y=-6
Step-by-step explanation:
y=-4(2)+2
y=-8+2
y=-6
(2,-6)
Answer:
y = - 6
Step-by-step explanation: You have to substitute 2 for x in order for you to get your solution.
y = - 4x + 2
y = - 4(2) + 2
y = - 8 + 2
y = - 6
Remember, a negative times a positive, becomes a negativeHope this helps you!!! :)
Jared bought a package of pens containing 20 pens if 3/4 of the pens have black ink how do you determine the number of pens with black ink
Answer:
The number of black ink pen are 15 pens.
Step-by-step explanation:
As given
Jared bought a package of pens containing 20 pens.
[tex]if\ \frac{3}{4}\ of\ the\ pens\ have\ black\ ink.[/tex]
Thus
[tex]Number\ of\ black\ ink\ pens = \frac{3}{4}\times Total\ number\ of\ pens[/tex]
Putting the value
[tex]Number\ of\ black\ ink\ pens = \frac{3}{4}\times 20[/tex]
Number of black ink pens = 3 × 5
= 15 pens
Therefore the number of black ink pen are 15 pens.
Jared has 15 pens with black ink, which is calculated by multiplying 3/4 by the total number of pens he has (20 pens).
To determine the number of pens with black ink that Jared has, you need to multiply the total number of pens by the fraction representing the pens with black ink. Jared has a package containing 20 pens, and 3/4 of these have black ink. Here's how you calculate it:
Find the fraction of pens with black ink: 3/4 of the pens.
Multiply this fraction by the total number of pens: 3/4 × 20 pens.
Calculate the product to find the number of pens with black ink: 3/4 × 20 = 15 pens.
Therefore, Jared has 15 pens that contain black ink.
A certain type of bacteria increases continuously at a rate proportional to the number present. if there are 500 present at a given time and 1,000 present 2 hours later, how many hours (from the initial given time) will it take for the numbers to be 2,500
The number of hours from the initial given time that will take for the bacteria to grow to 2500 number is 4.64 hours approx.
How to calculate the instantaneous rate of growth of a function?Suppose that a function is defined as;
[tex]y = f(x)[/tex]
Then, suppose that we want to know the instantaneous rate of the growth of the function with respect to the change in x, then its instantaneous rate is given as:
[tex]\dfrac{dy}{dx} = \dfrac{d(f(x))}{dx}[/tex]
Assuming that the bacterial growth can be approximated by a continous and differentiable function y = f(x), where x represents the number of hours spent from the initial time, we're given that:
[tex]\dfrac{dy}{dx} \propto y[/tex]Supposing the proportionality constant be k, then we get:
[tex]\dfrac{dy}{dx} = ky[/tex]
Solving this differential equation, we get:
[tex]\dfrac{dy}{y} = kdx\\\\\text{Integrating both the sides without limits}\\\\\int \dfrac{dy}{y} = \int x dx\\\\\ln(y) + \ln(c) = kx\\\\\ln(yc) =kx\\ yc = e^{kx}\\y = \dfrac{e^{kx}}{c}[/tex]
where ln(c) represents the integration constant. (we took ln(c) because, firstly, ln's range is whole real number (which gives us the access to use it as integration constant), and secondly that it can merge with ln(y) to simplify the work)
Since we're given that:
At x = t (for some value of t in hours), we're given that y = 500,
and for x = t+2, y = 1000,
so we get two equations as:
[tex]\\500 = \dfrac{e^{kt}}{c}\\\\1000 = \dfrac{e^{k(t+2)}}{c}\\[/tex]
Thus, we get:
[tex]\dfrac{e^{kt}}{500} = \dfrac{e^{k(t+2)}}{1000} \\\\kt = \ln(0.5) + k(t+2)\\\\k = \dfrac{-\ln(0.5)}{2}} \approx 0.3465[/tex]
Thus, we get:
[tex]\\500 = \dfrac{e^{kt}}{c} \\\\c = \dfrac{e^{0.3465t}}{500}[/tex]
Thus, we get:
[tex]y = \dfrac{e^{0.3465x}}{\dfrac{e^{0.3465t}}{500}} = 500e^{0.3465(x-t)[/tex]
Let from the initial given time t, it takes h hours more for bacterias to be 2500, then we get:
[tex]2500 = 500 \times e^{0.3465 (t+h - t)}\\0.3465(h) = \ln(5)\\\\h = \dfrac{\ln(5)}{0.3465} = 4.644 \: \rm hours \: approx.[/tex]
Thus, the number of hours from the initial given time that will take for the bacteria to grow to 2500 number is 4.64 hours approx.
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