Chapter 2.6, Problem 9AYU

9. A rectangle is inscribed in a semicircle of radius 2. See the figure. Let $P\text{ = }\left(x,y\right)$ be the point in quadrant I that is a vertex of the rectangle and is on the circle.

(a) Express the area A of the rectangle as a function of x.

(b) Express the perimeter p of the rectangle as a function of x.

(c) Graph $A\text{ = }A\left(x\right)$. For what value of x is A largest?

(d) Graph $\text{P = P}\left(x\right)$. For what value of x is p largest?

To determine

To find:

a. To express the area $A$ of the rectangle as a function of $x$.

Answer to Problem 9AYU

Solution:

a. $4x\sqrt{4-x^2}$

Explanation of Solution

Given:

$y=\sqrt{4-x^2}$

Calculation:

a. To express the area $A$ of the rectangle as a function of $x$:

Area of the given rectangle $=A=4xy$.

Since the rectangle is inscribed in the semi circle of radius 2. (i.e) $y=\sqrt{4-x^2}$. Also, $p$ be the point in quadrant I that is the vertex of the rectangle and it is on the circle.

Substitute $y=\sqrt{4-x^2}$ for $y$. Then

$A\left(x\right)=4x\sqrt{4-x^2}$

To determine

To find:

b. To express the perimeter $p$ of the rectangle as a function of $x$.

Answer to Problem 9AYU

Solution:

b. $4x+2\sqrt{4-x^2}$

Explanation of Solution

Given:

$y=\sqrt{4-x^2}$

Calculation:

b. To express the perimeter $p$ of the rectangle as a function of $x$:

Perimeter of the given rectangle $p=2\left(2 x+2y\right)=4\left(x+y\right)$.

Since the rectangle is inscribed in the semi circle of radius 2. (i.e) $y=\sqrt{4-x^2}$. Also, $P$ be the point in quadrant I that is the vertex of the rectangle and it is on the circle.

Substitute $y=\sqrt{4-x^2}$ for $y$. Then

$p=4\left(x+y\right)=4\left(x+\sqrt{4-x^2}\right)=4x+4\sqrt{4-x^2}$

To determine

To find:

c. To graph $A=A\left(x\right)$ and to find the value of $x$ is $A$ largest.

Answer to Problem 9AYU

Solution:

c. $A$ is largest when $x=1.41$.

Explanation of Solution

Given:

$y=\sqrt{4-x^2}$

Calculation:

c. To graph $A=A\left(x\right)$ and to find the value of $x$ is $A$ largest:

From the graph it can be seen that $A=7.99$ is largest when $x=1.41$.

To determine

To find:

d. To graph $p=p\left(x\right)$ ) and to find the value of $x$ is $p$ largest.

Answer to Problem 9AYU

Solution:

d. $p$ is largest when $x=1.41$.

Explanation of Solution

Given:

$y=\sqrt{4-x^2}$

Calculation:

d. To graph $p=p\left(x\right)$ and to find the value of $x$ is $p$ largest:

From the graph it can be seen that $p=11.31$ is largest when $x=1.41$.