The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled asp(x)0.00022x30.014x2- 0.0033x 12.236 percentwhere x is the number of years since 2000, data from 0 s x s 50.tDetermine the value of x in the domain 0x s 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places)Thus, in which year is the percentage is predicted to be increasing most rapidly?Calculate the percentage at that time. (Round your answer to two decimal places.)Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.)percentage points per year

Question
Asked Nov 19, 2019
1 views
The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as
p(x)0.00022x30.014x2- 0.0033x 12.236 percent
where x is the number of years since 2000, data from 0 s x s 50.t
Determine the value of x in the domain 0
x s 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places)
Thus, in which year is the percentage is predicted to be increasing most rapidly?
Calculate the percentage at that time. (Round your answer to two decimal places.)
Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.)
percentage points per year
help_outline

Image Transcriptionclose

The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as p(x)0.00022x30.014x2- 0.0033x 12.236 percent where x is the number of years since 2000, data from 0 s x s 50.t Determine the value of x in the domain 0 x s 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places) Thus, in which year is the percentage is predicted to be increasing most rapidly? Calculate the percentage at that time. (Round your answer to two decimal places.) Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.) percentage points per year

fullscreen
check_circle

Expert Answer

Step 1

First of all, differentiate p(x) w.r.t. x and get p '(x).

A derivative of a function shows the increasing or decreasing nature of the function.

Р(x) --0.00022х +0.014x? 0.0033к + 12.236
d
-0.00022x) +0.014x* - 0.0033х +12.236)
- -о
p'(x)
—-0.00066х- +0.028х —0.0033
help_outline

Image Transcriptionclose

Р(x) --0.00022х +0.014x? 0.0033к + 12.236 d -0.00022x) +0.014x* - 0.0033х +12.236) - -о p'(x) —-0.00066х- +0.028х —0.0033

fullscreen
Step 2

Now, again differentiate p '(x) w.r.t. x to get p '' (x) and equate it to 0 to get the critical points.

The critical point is at x = 21.212

p'(x)-0.00066x 0.028.r-0.0033
p"(x)=-(-0.00066x2 +0.028x - 0.0033)
=-0.00132.r + 0.028
p"(x)= 0
-0.00132x
0.028 = 0
-0.028
-0.00132
- 21.2121
21.212 (upto 3 decimals)
help_outline

Image Transcriptionclose

p'(x)-0.00066x 0.028.r-0.0033 p"(x)=-(-0.00066x2 +0.028x - 0.0033) =-0.00132.r + 0.028 p"(x)= 0 -0.00132x 0.028 = 0 -0.028 -0.00132 - 21.2121 21.212 (upto 3 decimals)

fullscreen
Step 3

Then find the value of p '' (x) at the critical point i.e. x = 21.212.

The value of p '' (x) at the critical point is 0.00000016 and...

p"(x)=-0.00132r+ 0.028
p"(21.212)-0.00132(21.212)+ 0.028
0.00000016 (+ve)
help_outline

Image Transcriptionclose

p"(x)=-0.00132r+ 0.028 p"(21.212)-0.00132(21.212)+ 0.028 0.00000016 (+ve)

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Calculus

Derivative

Related Calculus Q&A

Find answers to questions asked by student like you

Show more Q&A add
question_answer

Q: Can someone please show me the steps for solving this problem? The answer is 68 but I don't know how...

A: The given equation is

question_answer

Q: The graph ofg consists of two straight lines and a semicircle. Use it to evaluate each integral. У y...

A: Given graph is,

question_answer

Q: find the limit. use l'Hospital's rule where appropriate.

A: Refer to the question we need to find the limit of the following expression

question_answer

Q: A visitor is staying in a cottage that is 10 miles east of the closest point on a shoreline to an is...

A: Given A visitor is staying in a cottage that is 10 miles east of the closest point on a shoreline to...

question_answer

Q: Determine whether the following sequence converges or diverges. If it converges, then find its limit...

A: Given, sequence is

question_answer

Q: Use synthetic division to perform the indicated division. Write the polynomial in the form p(x)=d(x)...

A: The synthetic division (3x2-2x2+1)/(x-1) is performed as follows.

question_answer

Q: A rancher has 4,648 feet of fencing available to enclose a rectangular area bordering a river. He wa...

A: Let the figure of the fencing be:

question_answer

Q: Letf" () 8t 10Ví (a) Find the most general formula for f"(t). If an arbitrary constant must be used ...

A: Given:

question_answer

Q: The population of mice in Alfred is given by P(t)=2116e^(5t) , where t is in years since 1986. What ...

A: To find the rate of change we find the first derivative of P(t) using chain rule.Answer: Rate of cha...