opatovaL

2021-03-07

Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by $P=-0.00232{t}^{3}+0.0151{y}^{2}+2.83t+281.8,0\le t\le 13,$
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$P=-0.00232{t}^{3}+0.0151{y}^{2}+2.83t+281.8,0\le t\le 13$ over the closed interval [0, 13].

(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

brawnyN

Skilled2021-03-08Added 91 answers

Used formula:

The derivative of power function given by

$\frac{d}{dx}({x}^{n})=n{x}^{n-1}$

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$P=P=-0.00232{t}^{3}+0.0151{y}^{2}+2.83t+281.8,$

Step 1. Determine first derivative of the function$P=P=-0.00232{t}^{3}+0.0151{y}^{2}+2.83t+281.8,0$

$P\u2018=-0.0069{t}^{2}+0.03t+2.83$

Step 2. Find out the critical points of P. It occurs when$P\u2018=0$ or P` underfined.

$P\u2018=0$

$-0.0069{t}^{2}+0.03t+2.83=0$

Apply quadratix formula as,

$t=\frac{-0.03\pm {\sqrt{0.03}}^{2}-4(-0.0069)(2.83)}{2(-0.0069)}$

$=\frac{-0.03\pm \sqrt{0.0009+0.078108}}{-0.0138}$

$=\frac{-0.03\pm \sqrt{0.079008}}{-0.0138}$

$=\frac{-0.03\pm 0.281}{-0.0138}$

Further solving,

$t=\frac{-0.03\pm 0.281}{-0.0138}$

$=-18.19$

And,

$t=\frac{-0.03-0.281}{-0.0138}$

$=22.53$

This gives,

$t=-18.19,22.53$

For end-point$t=0,$

Substitute$t=0$
in the function $P=-0.00232{t}^{3}+0.0151{y}^{2}+2.83t+281.8,$

$P=-0.0023(0{)}^{3}+0.015(0{)}^{2}+2.83(0)+281.8,$

$=281.8$

For end-point$t=13,$

Substitute$t=13$
in the function $P=-{0.0023}^{3}+{0.015}^{2}+2.83t+281.8,$

$P=-0.0023(13{)}^{3}+0.015(13{)}^{2}+2.83(13)+281.8,$

$=-0.0023(2197)+0.015(169)+3.83(13)+281.8$

$=-5.531+2.535+36.79+281.8$

$=316.1$

Use all the information to form the table as,

$\begin{array}{|ccc|}\hline t-value& s=0& r=13\\ P& 281.8& 316.1\\ Conclusion& Minimum& Maximum\\ \hline\end{array}$

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.

The derivative of power function given by

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

Step 1. Determine first derivative of the function

Step 2. Find out the critical points of P. It occurs when

Apply quadratix formula as,

Further solving,

And,

This gives,

For end-point

Substitute

For end-point

Substitute

Use all the information to form the table as,

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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