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Chapter 2: Q. 8 (page 119)

A rectangle is inscribed in a semicircle of radius 2. See the figure. Let P=x,ybe 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 pof the rectangle as a function of x.

Part (c): Graph A=Ax. For what value of xis Alargest?

Part (d): Graph p=px. For what value of xis plargest?

Short Answer

Expert verified

Part (a): Area of the rectangle as a function is Ax=2x脳4-x2.

Part (b): Perimeter of the rectangle as a function is px=4x+24-x2.

Part (c): On plotting the function A=Ax, we get,

The value of x for which Ais the largest isx鈮�1.41.

Part (d): On plotting the function p=px, we get,

The value of x for which pis the largest isx鈮�1.79.

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given figure,

Distance between the points -x,0,x,0is the length of the rectangle,

l=x--x2+0-02=2x2=2x

02

Part (a) Step 2. Find the breadth.

Consider the given figure,

Distance between the points x,0,x,yis the breadth of the rectangle,

localid="1645804169041" b=x-x2+y-02=y2=y=4-x2

We know the area of the rectangle is A=l脳b.

Substitute the values,

Ax=2x脳4-x2

03

Part (b) Step 1. Given information.

We know the perimeter of the rectangle is p=2l+b.

Substitute the values,

px=22x+4-x2px=4x+24-x2

04

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

On plotting the function Ax=2x脳4-x2, we get,

From the graph, we can say that the largest value of xis largestAatx鈮�1.41.

05

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

On plotting the function px=4x+24-x2, we get,

From the graph, we can say that the largest value of xis largestpat x鈮�1.79.

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