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+281.8,0≤t≤13, 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+281.8,0≤t≤13 over the closed interval [0, 13].

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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+281.8,0≤t≤13, 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+281.8,0≤t≤13 over the closed interval [0, 13].

brawnyN · Accepted Answer

Used formula:  The derivative of power function given by  ddx(xn)=nxn−1  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±0.032−4(−0.0069)(2.83)2(−0.0069)  =−0.03±0.0009+0.078108−0.0138  =−0.03±0.079008−0.0138  =−0.03±0.281−0.0138  Further solving,  t=−0.03±0.281−0.0138  =−18.19  And,  t=−0.03−0.281−0.0138  =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,  t−values=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.

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

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