Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by P = -0.00232t^{3} + 0.0151y^{2} + 2.8

opatovaL 2021-03-07 Answered
Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by P=0.00232t3+0.0151y2+2.83t+281.8,0t13, where t=0 corresponds to 2000.
(Source: U.S. Census Bureau)
Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013.
Analytically find the maximum and minimum populations over the interval.
The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
We need to calculate: The absolute extrema of the popullation P=0.00232t3+0.0151y2+2.83t+281.8,0t13 over the closed interval [0, 13].
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Expert Answer

brawnyN
Answered 2021-03-08 Author has 91 answers
Used formula:
The derivative of power function given by
ddx(xn)=nxn1
Procedure to find the extrema of the continuous function f on closed interval [a, b].
Step 1: Find the derivative of the function f.
Step 2: Find the critical points of f in the open interval (a, b).
Step 3: Determine the value of f at each of the critical numbers in the open interval (a,b).
Step 4: Determine the value of f at each of the end-points a and b.
Step 5: The least of these values is the minimum and the greatest is the maximum.
Calculation:
Consider the function P=P=0.00232t3+0.0151y2+2.83t+281.8,
Step 1. Determine first derivative of the function P=P=0.00232t3+0.0151y2+2.83t+281.8,0
P=0.0069t2+0.03t+2.83
Step 2. Find out the critical points of P. It occurs when P=0 or P` underfined.
P=0
0.0069t2+0.03t+2.83=0
Apply quadratix formula as,
t=0.03±0.0324(0.0069)(2.83)2(0.0069)
=0.03±0.0009+0.0781080.0138
=0.03±0.0790080.0138
=0.03±0.2810.0138
Further solving,
t=0.03±0.2810.0138
=18.19
And,
t=0.030.2810.0138
=22.53
This gives,
t=18.19,22.53
For end-point t=0,
Substitute t=0 in the function P=0.00232t3+0.0151y2+2.83t+281.8,
P=0.0023(0)3+0.015(0)2+2.83(0)+281.8,
=281.8
For end-point t=13,
Substitute t=13 in the function P=0.00233+0.0152+2.83t+281.8,
P=0.0023(13)3+0.015(13)2+2.83(13)+281.8,
=0.0023(2197)+0.015(169)+3.83(13)+281.8
=5.531+2.535+36.79+281.8
=316.1
Use all the information to form the table as,
tvalues=0r=13P281.8316.1ConclusionMinimumMaximum
From the table, it can be concluded that the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
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