Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

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

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

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

Part (c): Graph $A=A\left(x\right)$. For what value of x is A largest?

Part (d): Graph $p=p\left(x\right)$. For what value of x is p largest?

Step-by-Step Solution

Answer

Part (a): Area of the rectangle as a function is $A\left(x\right)=2x\times \sqrt{4-x^2}$.

Part (b): Perimeter of the rectangle as a function is $p\left(x\right)=4x+2\sqrt{4-x^2}$.

Part (c): On plotting the function $A=A\left(x\right)$, we get,

The value of x  for which A is the largest is $x\approx 1.41$.

Part (d): On plotting the function $p=p\left(x\right)$, we get,

The value of x  for which p is the largest is $x\approx 1.79$.

Part (a) Step 1. Given information.

Consider the given figure,

Distance between the points $\left(-x,0\right),\left(x,0\right)$ is the length of the rectangle,

$$\begin{gathered} l=\sqrt{{\left\{x-\left(-x\right)\right\}}^2+{\left(0-0\right)}^2} \\ =\sqrt{{\left(2x\right)}^2} \\ =2x \end{gathered}$$

Part (a) Step 2. Find the breadth.

Consider the given figure,

Distance between the points $\left(x,0\right),\left(x,y\right)$ is the breadth of the rectangle,

$$\begin{gathered} b=\sqrt{{\left(x-x\right)}^2+{\left(y-0\right)}^2} \\ =\sqrt{y^2} \\ =y \\ =\sqrt{4-x^2} \end{gathered}$$

We know the area of the rectangle is $A=l\times b$.

Substitute the values,

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

Part (b) Step 1. Given information.

We know the perimeter of the rectangle is $p=2\left(l+b\right)$.

Substitute the values,

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

Part (c) Step 1. Plot the function of A = A x .

On plotting the function $A\left(x\right)=2x\times \sqrt{4-x^2}$, we get,

From the graph, we can say that the largest value of x is largest A at $x\approx 1.41$.

Part (d) Step 1. Plot the function of p = p x .

On plotting the function $p\left(x\right)=4x+2\sqrt{4-x^2}$, we get,

From the graph, we can say that the largest value of x is largest p at $x\approx 1.79$.