Answer:
a. L{t} = 1/s² b. L{1} = 1/s
Step-by-step explanation:
Here is the complete question
The The Laplace Transform of a function ft), which is defined for all t2 0, is denoted by Lf(t)) and is defined by the improper integral Lf))s)J" e-st . f(C)dt, as long as it converges. Laplace Transform is very useful in physics and engineering for solving certain linear ordinary differential equations. (Hint: think of s as a fixed constant) 1. Find Lft) (hint: remember integration by parts) A. None of these. B. O C. D. 1 E. F. -s2 2. Find L(1) A. 1 B. None of these. C. 1 D.-s E. 0
Solution
a. L{t}
L{t} = ∫₀⁰⁰[tex]e^{-st}t[/tex]
Integrating by parts ∫udv/dt = uv - ∫vdu/dt where u = t and dv/dt = [tex]e^{-st}[/tex] and v = [tex]\frac{e^{-st}}{-s}[/tex] and du/dt = dt/dt = 1
So, ∫₀⁰⁰udv/dt = uv - ∫₀⁰⁰vdu/dt w
So, ∫₀⁰⁰[tex]e^{-st}t[/tex] = [[tex]\frac{te^{-st}}{-s}[/tex]]₀⁰⁰ - ∫₀⁰⁰ [tex]\frac{e^{-st}}{-s}[/tex]
∫₀⁰⁰[tex]e^{-st}t[/tex] = [[tex]\frac{te^{-st}}{-s}[/tex]]₀⁰⁰ - ∫₀⁰⁰ [tex]\frac{e^{-st}}{-s}[/tex]
= -1/s(∞exp(-∞s) - 0 × exp(-0s)) + [tex]\frac{1}{s}[/tex] [[tex]\frac{e^{-st} }{-s}[/tex]]₀⁰⁰
= -1/s[(∞exp(-∞) - 0 × exp(0)] - 1/s²[exp(-∞s) - exp(-0s)]
= -1/s[(∞ × 0 - 0 × 1] - 1/s²[exp(-∞) - exp(-0)]
= -1/s[(0 - 0] - 1/s²[0 - 1]
= -1/s[(0] - 1/s²[- 1]
= 0 + 1/s²
= 1/s²
L{t} = 1/s²
b. L{1}
L{1} = ∫₀⁰⁰[tex]e^{-st}1[/tex]
= [[tex]\frac{e^{-st} }{-s}[/tex]]₀⁰⁰
= -1/s[exp(-∞s) - exp(-0s)]
= -1/s[exp(-∞) - exp(-0)]
= -1/s[0 - 1]
= -1/s(-1)
= 1/s
L{1} = 1/s
Given coordinates A(3,3),B(2,5),C(4,3) complete transformation. Complete double reflection over the lines y=2 followed by y=0.
9514 1404 393
Answer:
A"(3, -1)B"(2, 1)C"(4, -1)Step-by-step explanation:
Reflection over 'a' then over 'b' will result in a translation of 2(b -a). Here, we have a=2, b=0, so the translation is 2(0-2) = -4. The reflection is over horizontal lines, so the transformation is ...
(x, y) ⇒ (x, y -4)
A(3, 3) ⇒ A"(3, -1)
B(2, 5) ⇒ B"(2, 1)
C(4,3) ⇒ C"(4, -1)
Find the missing Side of the triangle
Answer:
2√15
Step-by-step explanation:
Use the Pythagorean theorem.
2² + x² = 8²
x² + 4 = 64
x² = 60
x² = 4 * 15
x = 2√15
Solve the equation.
1. For parentheses:
Distribute
4-2(x+7) = 3(x+5)
2. If necessary:
Combine Terms
3. Apply properties:
Add Subtract
Multiply
Divide
4. To start over:
Reset
Answer:
x = -5
Step-by-step explanation:
4-2(x+7) = 3(x+5)
Distribute
4 - 2x-14 = 3x+15
Combine like terms
-2x-10 = 3x+15
Add 2x to each side
-2x-10 +2x =3x+2x+15
-10 = 5x+15
Subtract 15 from each side
-10-15 = 5x+15-15
-25 = 5x
Divide by 5
-25/5 = 5x/5
-5 =x
Please look below (Please Explain and NO LINKS)
Answer:
Mean = Sum of all numbers divided by the amount of numbers
[tex]Mean/Average=\frac{3+1+1.5+1.25+2.25+4+1+2}{8} =\frac{16}{8} =2[/tex]
Median = the middle number when the ordered from least to greatest.
From least to greatest: [tex]1, 1, 1.25, 1.5, 2, 2.25, 3, 4[/tex]The two middle numbers are 1.5 and 2.If there are two middle numbers, find the mean/average of those numbers:
[tex]\frac{1.5+2}{2} =\frac{3.5}{2} =1.75[/tex]
Therefore, the answer would be:
Mean = 2Median = 1.75Instructions are in the picture
Answer:
123123 3213123 12312 dasdsd aw dasd sda asdasd
Step-by-step explanation:
What is the value of |-6|—|6|-(-6)?
The solution is
Answer:
6
Step-by-step explanation:
|-6| = 6
|6| = 6
- -6 = +6
so, we have
6 - 6 + 6 = 6
At a time hours after taking a tablet, the rate at which a drug is being eliminated r(t)= 50 (e^-01t - e^-0.20t)is mg/hr. Assuming that all the drug is eventually eliminated, calculate the original dose.
Answer:
2500 mg
Step-by-step explanation:
Since r(t) is the rate at which the drug is being eliminated, we integrate r(t) with t from 0 to ∞ to find the original dose of drug, m. Since all of the drug will be eliminated at time t = ∞.
Since r(t) = 50 (e^-01t - e^-0.20t)
m = ∫₀⁰⁰50 (e^-01t - e^-0.20t)
= 50∫₀⁰⁰(e^-01t - e^-0.20t)
= 50[∫₀⁰⁰e^-01t - ∫₀⁰⁰e^-0.20t]
= 50([e^-01t/-0.01]₀⁰⁰ - [e^-0.20t/-0.02]₀⁰⁰)
= 50(1/-0.01[e^-01(∞) - e^-01(0)] - {1/-0.02[e^-0.02(∞) - e^-0.02(0)]})
= 50(1/-0.01[e^-(∞) - e^-(0)] - {1/-0.02[e^-(∞) - e^-(0)]})
= 50(1/-0.01[0 - 1] - {1/-0.02[0 - 1]})
= 50(1/-0.01[- 1] - {1/-0.02[- 1]})
= 50(1/0.01 - 1/0.02)
= 50(100 - 50)
= 50(50)
= 2500 mg
in the figure above, the square ABCD is inscribed in a circle. if the radius of the circle is r, the hatbis the length of arc APD in terms of r?
a) (pi)r/4
b) (pi)r/2
c) (pi)r
d) (pi)r^2/4
The length of arc APD is: [tex]\frac{\pi r}{2}[/tex]
A square when inscribed in a circle will fit the circle such that, the 4 edges of the square touches the sides of the circle. The radius of the circle can be drawn from any of the 4 edges.
Given that ABCD is a square:
This means that:
[tex]AB = BC = CD = DA[/tex] --- equal side lengths
To calculate the length of arc APD, we make use of the following arc length formula
[tex]APD = \frac{\theta}{360} * 2\pi r[/tex]
Where
[tex]\theta = \angle ADO[/tex] and O is circle center
Since ABCD is a square, then:
[tex]\theta = \angle ADO = 90^o[/tex]
So, we have:
[tex]APD = \frac{90}{360} * 2\pi r[/tex]
[tex]APD = \frac{1}{4} * 2\pi r[/tex]
[tex]APD = \frac{\pi r}{2}[/tex]
Read more at:
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What are four ways an inequality can be written?
Answer:
There are four ways to represent an inequality: Equation notation, set notation, interval notation, and solution graph.
Eli takes the 17 apples home, and he bakes as many apple pies
as he can. He uses 7 apples in each pie. How many apple pies does
Eli bake? How many apples are left?
Answer:
2 with 3 left over
Step-by-step explanation:
17 divided by 2 is 14 with 3 remaining
Answer:
2 pies
Step-by-step explanation:
Can someone do #2?❤️
Answer:
b
Step-by-step explanation:
A proportional relationship is a straight line. Is must also go through the point (0,0)
b
Answer:
Step-by-step explanation:
A proportional relationship is a straight line. Is must also go through the point (0,0)
What is the value of b? -11b + 7 =40 (also there is another question in the bottom of the picture. If you can answer it please do)
Problem 1
The idea here is to follow PEMDAS in reverse to undo what is happening to the variable b, so we can isolate it.
-11b + 7 = 40
-11b = 40-7
-11b = 33
b = 33/(-11)
b = -3
To check this value, plug it back into the original equation. You should get 40 on each side to help confirm the answer.
Answer: b = -3=====================================================
Problem 2
There are two ways we can solve. One method is to use the hint your teacher gave you. So we'll distribute first and then follow the same idea as problem 1
9(p-4) = -18
9p-36 = -18
9p = -18+36
9p = 18
p = 18/9
p = 2
Another method you can use is to follow these steps
9(p-4) = -18
p-4 = -18/9
p-4 = -2
p = -2+4
p = 2
Either way, we get the same result. To check the answer, replace every p with 2 in the original equation. You should get -18 on the left side after simplifying.
Answer: p = 2Translate into an algebraic expression:
n-1 increased by 110%
Answer:
Step-by-step explanation:
(n-1)1.1
A car rental agency rents 480 cars per day at a rate of $20 per day. For each $1 increase in rate, 10 fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?
Answer:
340 cars at $ 34 should be rented to produce the maximum income of $ 11,560.
Step-by-step explanation:
Given that a car rental agency rents 480 cars per day at a rate of $ 20 per day, and for each $ 1 increase in rate, 10 fewer cars are rented, to determine at what rate should the cars be rented to produce the maximum income and what is the maximum income, the following calculations must be performed:
480 x 20 = 9600
400 x 28 = 11200
350 x 33 = 11550
300 x 38 = 11400
310 x 37 = 11470
320 x 36 = 11520
330 x 35 = 11550
340 x 34 = 11560
Therefore, 340 cars at $ 34 should be rented to produce the maximum income of $ 11,560.
The length of a rectangular field is 25 m more than its width. The perimeter of the field is 450 m. What is the actual width and length?
Answer:
length= 125
width= 100
Step-by-step explanation:
let width have a length of x m
therefore length= (x+25)m
perimeter=2(length +width)
p=2((x+25)+x)
p=4x+50
but we have perimeter to be 450,, we equate it to 4x+50 above,
450=4x+50
4x=400
x=100 m
length= 125
width= 100
If sum of first 6 digits of AP is 36 and that of the first 16 terms is 255,then find the sum of first ten terms.
•Please answer it correctly ( step by step)
Answer:
100
Step-by-step explanation:
We have the sum of first n terms of an AP,
Sn = n/2 [2a+(n−1)d]
Given,
36= 6/2 [2a+(6−1)d]
12=2a+5d ---------(1)
256= 16/2 [2a+(16−1)d]
32=2a+15d ---------(2)
Subtracting, (1) from (2)
32−12=2a+15d−(2a+5d)
20=10d ⟹d=2
Substituting for d in (1),
12=2a+5(2)=2(a+5)
6=a+5 ⟹a=1
∴ The sum of first 10 terms of an AP,
S10 = 10/2 [2(1)+(10−1)2]
S10 =5[2+18]
S10 =100
This is the sum of the first 10 terms.
Hope it will help.
[tex]\sf\underline{\underline{Question:}}[/tex]
If sum of first 6 digits of AP is 36 and that of the first 16 terms is 255,then find the sum of first ten terms.
$\sf\underline{\underline{Solution:}}$
$\sf\bold\purple{||100||}$$\space$
$\sf\underline\bold\red{||Step-by-Step||}$
$\sf\bold{Given:}$
$\sf\bold{S6=36}$ $\sf\bold{S16=255}$$\space$
$\sf\bold{To\:find:}$
$\sf\bold{The \: sum\:of\:the\:first\:ten\:numbers}$$\space$
$\sf\bold{Formula\:we\:are\:using:}$
$\implies$ $\sf{ Sn=}$ $\sf\dfrac{N}{2}$ $\sf\small{[2a+(n-1)d]}$
$\space$
$\sf\bold{Substituting\:the\:values:}$
→ $\sf{S6=}$ $\sf\dfrac{6}{2}$ $\sf\small{[2a+(6-1)d]}$
→ $\sf{36 = 3[2a+(6-1)d]}$
→$\sf{12=[2a+5d]}$ $\sf\bold\purple{(First \: equation)}$
$\space$
$\sf\bold{Again,Substituting \: the\:values:}$
→ $\sf{S16}$ $\sf\dfrac{16}{2}$ $\sf\small{[2a+(16-1)d]}$
→ $\sf{255=8[2a + (16-1)d]}$
:: $\sf\dfrac{255}{8}$ $\sf\small{=31.89=32}$
→ $\sf{32=[2a+15d]}$ $\sf\bold\purple{(Second\:equation)}$
$\space$
$\sf\bold{Now,Solve \: equation \: 1 \:and \:2:}$
→ $\sf{10=20}$
→ $\sf{d=}$ $\sf\dfrac{20}{10}$ $\sf{=2}$
$\space$
$\sf\bold{Putting \: d=2\: in \:equation - 1:}$
→ $\sf{12=2a+5\times 2}$
→ $\sf{a = 1}$
$\space$
$\sf\bold{All\:of\:the\:above\:eq\: In \: S10\:formula:}$
$\mapsto$ $\sf{S10=}$ $\sf\dfrac{10}{2}$ $\sf\small{[2\times1+(10-1)d]}$
$\mapsto$ $\sf{5(2\times1+9\times2)}$
$\mapsto$ $\sf\bold\purple{5(2+18)=100}$
$\space$
$\sf\small\red{||Hence , the \: sum\: of \: the \: first\:10\: terms\: is\:100||}$
_____________________________
A researcher records the repair cost for 27 randomly selected refrigerators. A sample mean of $60.52 and standard deviation of $23.29 are subsequently computed. Determine the 90% confidence interval for the mean repair cost for the refrigerators. Assume the population is approximately normal. Step 1 of 2 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.
Answer:
The critical value is [tex]T_c = 1.7056[/tex]
The 90% confidence interval for the mean repair cost for the refrigerators is ($52.875, $68.165).
Step-by-step explanation:
We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.
The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So
df = 27 - 1 = 26
90% confidence interval
Now, we have to find a value of T, which is found looking at the t table, with 26 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.9}{2} = 0.95[/tex]. So we have T = 1.7056, which means that the critical value is [tex]T_c = 1.7056[/tex]
The margin of error is:
[tex]M = T\frac{s}{\sqrt{n}} = 1.7056\frac{23.29}{\sqrt{27}} = 7.645[/tex]
In which s is the standard deviation of the sample and n is the size of the sample.
The lower end of the interval is the sample mean subtracted by M. So it is 60.52 - 7.645 = $52.875.
The upper end of the interval is the sample mean added to M. So it is 60.52 + 7.645 = $68.165.
The 90% confidence interval for the mean repair cost for the refrigerators is ($52.875, $68.165).
Choose the correct elements in the set for the following:
{y | y is an integer and y >/= -3}
{3, 4, 5, 6, . . .}
{−2, −1, 0, 2, . . .}
{−1, 0, 1, 2, . . }
{−3, −2, −1, 0, . . .}
****PLEASE explain your answer****
Answer:
D
Step-by-step explanation:
Y => - 3 that is {−3, −2, −1, 0, . . .}
Assume that human body temperatures are normally distributed with a mean of 98.19 and a standard deviation of 0.61
Answer:
Ok I'm assuming that know what??
Step-by-step explanation:
A parallel plate capacitor has an area of 1.5 cm
2
and the plates are separated a distance of 2.0 mm with air between them. How much charge does this capacitor store when connected to a 12V battery?
Step-by-step explanation:
Given:
[tex]A=1.5\:\text{cm}^2×\left(\frac{1\:\text{m}^2}{10^4\:\text{cm}^2}\right)=1.5×10^{-4}\:\text{m}^2[/tex]
[tex]d = 2.0\:\text{mm} = 2.0×10^{-3}\:\text{mm}[/tex]
The charge stored in a capacitor is given by [tex]Q = CV.[/tex] In the case of a parallel-plate capacitor, its capacitance C is given by
[tex]C = \epsilon_0\dfrac{A}{d}[/tex]
where [tex]\epsilon_0[/tex] = permittivity of free space. The amount of charge stored in the capacitor is then
[tex]Q = \left(\epsilon_0\dfrac{A}{d}\right)V[/tex]
[tex]\:\:\:\:\:=\left[\dfrac{(8.85×10^{-12}\:\text{F/m})(1.5×10^{-4}\:\text{m}^2)}{(2.0×10^{-3}\:\text{m})}\right](12\:\text{V})[/tex]
[tex]\:\:\:\:\:=8.0×10^{-12}\:\text{C}[/tex]
helppppp plsss ??? plssss ??
Answer:
3 is correct dear i hope it will help uWhat is the extreme value of the polynomial function f(x)= x2 - 4?
Answer:
+∞.
Step-by-step explanation:
That would be positive infinity.
The extreme value of the given polynomial [tex]f(x) = x^{2} -4[/tex] is ∞.
What is extreme value of a polynomial?Extreme values of a polynomial are the peaks and valleys of the polynomial—the points where direction changes.
What are the steps of finding the extreme value of any polynomial?The following steps which are required to find the extreme value of polynomial are:
Arrange the polynomial into the the form of [tex]ax^{2} +bs+c[/tex] where a, b and c are numbers.Determine whether a, the coefficient of the [tex]x^{2}[/tex] term, is positive or negative.If the term is positive, the extreme value will be the infinity because the value will continue to grow as x increases.If it is negative, use the formula [tex]\frac{-b}{2a}[/tex] to find the value for extreme. And then plug [tex]x = \frac{-b}{2a}[/tex] in the original polynomial to calculate the extreme value of the polynomial.According to the given question.
We have a polynomial
[tex]f(x) = x^{2} -4[/tex]
Since, in the given polynomial the coefficient of [tex]x^{2}[/tex] is positive . Therefore, the extreme value of the given polynomial is infinity because the value will continue to grow as x increases.
Hence, the extreme value of the given polynomial [tex]f(x) = x^{2} -4[/tex] is ∞.
Find out more information about extreme value of a polynomial here:
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Find the remainder when f(x)=x3−4x2−6x−3 f ( x ) = x 3 − 4 x 2 − 6 x − 3 is divided by x+1
Answer:
The remainder is -2.
Step-by-step explanation:
According to the Polynomial Remainder Theorem, if we divide a polynomial P(x) by a binomial (x - a), then the remainder of the operation will be given by P(a).
Our polynomial is:
[tex]P(x) = x^3-4x^2-6x-3[/tex]
And we want to find the remainder when it's divided by the binomial:
[tex]x+1[/tex]
We can rewrite our divisor as (x - (-1)). Hence, a = -1.
Then by the PRT, the remainder will be:
[tex]\displaystyle\begin{aligned} R &= P(-1)\\ &=(-1)^3-4(-1)^2-6(-1)-3 \\ &= (-1)-4(1)+(6)-3 \\ &= -2 \end{aligned}[/tex]
The remainder is -2.
Do number 6 plz thanks
Answer:
24cm
Step-by-step explanation:
Question: Find the length of side OR.
Answer + explanation:
24cm
Since PQ = 24 cm, OR = 24 cm because they're paralleled and congruent!
Answer:
<O = 125
OR = 24
Step-by-step explanation:
consecutive angles are supplementary in a parallelogram
<R + <O = 180
55 + <O =180
<O = 180-55
< O = 125
opposite sides are congruent in a parallelogram
PQ = OR = 24
The physical plant at the main campus of a large state university recieves daily requests to replace fluorescent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 45 and a standard deviation of 3. Using the empirical rule, what is the approximate percentage of lightbulb replacement requests numbering between 42 and 45?
Do not enter the percent symbol.
ans = %
Answer:
34%
Step-by-step explanation:
Given that the distribution of daily light bulb request replacement is approximately bell shaped with ;
Mean , μ = 45 ; standard deviation, σ = 3
Using the empirical formula where ;
68% of the distribution is within 1 standard deviation from the mean ;
95% of the distribution is within 2 standard deviation from the mean
Lightbulb replacement numbering between ;
42 and 45
Number of standard deviations from the mean /
Z = (x - μ) / σ
(x - μ) / σ < Z < (x - μ) / σ
(42 - 45) / 3 = -1
This lies between - 1 standard deviation a d the mean :
Hence, the approximate percentage is : 68% / 2 = 34%
The consumer price index (CPI), issued by the U.S. Bureau of Labor Statistics, provides a means of determining the purchasing power of the U.S. dollar from one year to the next. Using the period from 1982 to 1984 as a measure of 100.0, the CPI figures for selected years from 2002 to 2016 are shown here. Year Consumer Price Index 2002 179.9 2004 188.9 2006 201.6 2008 215.3 2010 218.1 2012 229.6 2014 236.7 2016 240.0 E. To use the CPI to predict a price in a particular year, we can set up a proportion and compare it with a known price in another year, as follows. price in year A index in year A price in year B index in year B
In a model, a submarine is located at point (0, 0) on the coordinate plane. The submarine’s radar range has an equation of 2x2 + 2y2 = 128
Draw the figure on a graph and label the location of the submarine. Make sure your name is on the paper, and label this activity Part 2.
Can the submarine’s radar detect a ship located at the point (6, 6) ? Mark that location on your graph, and explain how you know whether or not the ship will be detected in the space provided on the Circles Portfolio Worksheet.
Answer:
Remember that for a circle centered in the point (a, b) and with a radius R, the equation is:
(x - a)^2 + (y - b)^2 = R^2
Here we know that the submarine is located at the point (0, 0)
And the radar range has the equation:
2*x^2 + 2*y^2 = 128
You can see that this seems like a circle equation.
If we divide both sides by 2, we get:
x^2 + y^2 = 128/2
x^2 + y^2 = 64 = 8^2
This is the equation for a circle centered in the point (0, 0) (which is the position of the submarine) of radius R = 8 units.
The graph can be seen below, this is just a circle of radius 8.
We also want to see if the submarine's radar can detect a ship located in the point (6, 6)
In the graph, this point is graphed, and you can see that it is outside the circle.
This means that it is outside the range of the radar, thus the radar can not detect the ship.
Find the measures of angles 1 and 2. If necessary, round to the tenths place.
Hint: Do not assume that Point D is the center of the circle.
A. m<1 = 20 m<2= 20
B. m<1 =40 m<2 = 140
C. m<1 = 82.5 m<2 = 97.5
D. m<1 =97.5 m<2= 82.5
Answer:
Option C
Step-by-step explanation:
From the picture attached,
m∠ABC = 40° [Given]
Since, measure of the intercepted arc is double of the measure of the inscribed angle.
Therefore, m(arc AC) = 2(m∠ABC)
m(arc AC) = 2(40°)
= 80°
m(arc FB) = 115° [Given]
By applying theorem of the angles formed by the chords inside a circle,
m∠2 = [tex]\frac{1}{2}(\text{arc}AC+\text{arc}FB)[/tex]
= [tex]\frac{1}{2}(80^{\circ}+115^{\circ})[/tex]
= 97.5°
m∠1 + m∠2 = 180° [Linear pair of angles are supplementary]
m∠1 + 97.5° = 180°
m∠1 = 180° - 97.5°
= 82.5°
Option C is the answer.
The population of Americans age 55 and older as a percentage of the total population is approximated by the function f(t) = 10.72(0.9t + 10)^0.3 (0 <= t < = 20)
where t is measured in years, with t=0 corresponding to the year 2000.
Required:
a. At what rate was the percentage of Americans age 55 and older changing at the beginning of 2002?
b. At what rate will the percentage of Americans age 55 and older be changing in 2017?
c. What will be the percentage of the population of Americans age 55 and older in 2017?
Answer:
Part A)
About 0.51% per year.
Part B)
About 0.30% per year.
Part C)
About 28.26%.
Step-by-step explanation:
We are given that the population of Americans age 55 and older as a percentange of the total population is approximated by the function:
[tex]f(t) = 10.72(0.9t+10)^{0.3}\text{ where } 0 \leq t \leq 20[/tex]
Where t is measured in years with t = 0 being the year 2000.
Part A)
Recall that the rate of change of a function at a point is given by its derivative. Thus, find the derivative of our function:
[tex]\displaystyle f'(t) = \frac{d}{dt} \left[ 10.72\left(0.9t+10\right)^{0.3}\right][/tex]
Rewrite:
[tex]\displaystyle f'(t) = 10.72\frac{d}{dt} \left[(0.9t+10)^{0.3}\right][/tex]
We can use the chain rule. Recall that:
[tex]\displaystyle \frac{d}{dx} [u(v(x))] = u'(v(x)) \cdot v'(x)[/tex]
Let:
[tex]\displaystyle u(t) = t^{0.3}\text{ and } v(t) = 0.9t+10 \text{ (so } u(v(t)) = (0.9t+10)^{0.3}\text{)}[/tex]
Then from the Power Rule:
[tex]\displaystyle u'(t) = 0.3t^{-0.7}\text{ and } v'(t) = 0.9[/tex]
Thus:
[tex]\displaystyle \frac{d}{dt}\left[(0.9t+10)^{0.3}\right]= 0.3(0.9t+10)^{-0.7}\cdot 0.9[/tex]
Substitute:
[tex]\displaystyle f'(t) = 10.72\left( 0.3(0.9t+10)^{-0.7}\cdot 0.9 \right)[/tex]
And simplify:
[tex]\displaystyle f'(t) = 2.8944(0.9t+10)^{-0.7}[/tex]
For 2002, t = 2. Then the rate at which the percentage is changing will be:
[tex]\displaystyle f'(2) = 2.8944(0.9(2)+10)^{-0.7} = 0.5143...\approx 0.51[/tex]
Contextually, this means the percentage is increasing by about 0.51% per year.
Part B)
Evaluate f'(t) when t = 17. This yields:
[tex]\displaystyle f'(17) = 2.8944(0.9(17)+10)^{-0.7} =0.3015...\approx 0.30[/tex]
Contextually, this means the percetange is increasing by about 0.30% per year.
Part C)
For this question, we will simply use the original function since it outputs the percentage of the American population 55 and older. Thus, evaluate f(t) when t = 17:
[tex]\displaystyle f(17) = 10.72(0.9(17)+10)^{0.3}=28.2573...\approx 28.26[/tex]
So, about 28.26% of the American population in 2017 are age 55 and older.
write 342 to 1 significant figure
Answer:
300
Step-by-step explanation:
A significant figure is the most important (largest) number you can round it to.
As it wants 1 significant figure, you count 1 to the left and round the 4 down.
Hope this helps :)
