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Chapter 2: Q. 7 (page 119)

A rectangle has one corner in quadrant I on the graph of y=16-x2, another at the origin, a third on the positive y-axis and the fourth on the positive x-axis. See the figure.

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

Part (b): What is the domain of A?

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

Short Answer

Expert verified

Part (a): The area A of the rectangle as a function is Ax=x16-x2.

Part (b): The domain of Ais 0,4.

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

The value of x for which Ais the largest isx鈮�2.31.

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given figure,

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

role="math" localid="1645801066135" l=x-02+0-02=x2=x

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,

b=x-x2+y-02=y2=y=16-x2

We know the area of the rectangle is localid="1645804194169" A=b脳l.

Substitute the values,

localid="1645802528646" Ax=x16-x2

03

Part (b) Step 1. Given information.

Consider the given question,

Area of the given rectangle is x16-x2.

As A>0, then x>0and role="math" localid="1645802654844" x<4.

Therefore, the domain is0,4.

04

Part (c) Step 1. Plot the function.

On plotting the function, we get,

From the graph, we can say that the largest value of x is largestAatx鈮�2.31

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