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Chapter 2: Q. 11 (page 120)

A wire 10meters long is to be cut into two pieces. One piece will be shaped as a square, and the other piece will be shaped as a circle. See the figure.

Part (a): Express the total area Aenclosed by the pieces of wire as a function of the length x of a side of the square.

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

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

Short Answer

Expert verified

Part (a): Total area Aenclosed by the pieces of wire as a function is Ax=x2+25-20x+4x2蟺.

Part (b): The domain of A is 0,52.

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

The value of x for which Ais smallest atx鈮�1.4m.

Step by step solution

01

Part (a) Step 1. Given information.

Assume x m to be the length of side of the square.

Perimeterp=4x

From the given figure,

Circumference of the circle formed is 10-4xm.

Assume r(x)to be the radius of the circle formed. Then it can be written,

localid="1645810318472" 2蟺谤x=10-4xrx=10-4x2蟺rx=5-2x蟺

02

Part (a) Step 2. Calculate the total area.

Consider the given question,

Ax=x2+蟺谤2Ax=x2+蟺5-2x蟺2Ax=x2+25-20x+4x2蟺

03

Part (b) Step 1. Find the domain of A.

Consider the given question,

rx=5-2x蟺. Also rx>0. Then,

5-2x蟺>05-2x>0x<52

x cannot be negative. As length cannot be negative.

Therefore, the domain is 0,52.

04

Part (c) Step 1. Plot the function.

On plotting the function, we get,

From the graph, we can say that Ais smallest at x鈮�1.4m.

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Most popular questions from this chapter

A wire 10meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle.

Part (a): Express the total area Aenclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle.

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

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

Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. Verify your results using a graphing utility.

g(x)=x3+1

If f(x)=x3if-2鈮�x<13x+2if1鈮�x鈮�4

Find: (a)f(-1) (b)f(0) (c)f(1) (d)f(3)

True or False The domain of (f.g)(x)consists of the numbers x that are in the domains of both fandg.

A circle of radius ris in inscribed in a square. See the figure.

Part (a): Express the area A of the square as a function of the radius rof the circle.

Part (b): Express the perimeter p of the square as a function of the radius rof the circle.
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