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Chapter 2: Q. 12 (page 120)

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?

Short Answer

Expert verified

Part (a): Total area Aenclosed by the pieces of wire as a function is Ax=34x2+100-60x+9x24蟺.

Part (b): The domain of A is localid="1645812756763" 0,103.

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

A is smallest atx鈮�2.08.

Step by step solution

01

Part (a) Step 1. Given information.

Assume xm to be the length of the given equilateral triangle.

Perimeterp=3xm

From the given figure,

Circumference of the circle formed is 10-3xm.

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

2蟺谤x=10-3xrx=10-3x2蟺

02

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

Consider the given question,

Total area A(x) enclosed by the pieces of wire is the sum of the areas enclosed by the triangle and circle,

Ax=34x2+蟺谤2Ax=34x2+蟺10-3x2蟺2Ax=34x2+100-60x+9x24蟺

03

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

Consider the given question,

rx=10-3x2蟺. Also rx>0. Then,

10-3x2蟺>010-3x>0x<103

x cannot be negative. As length cannot be negative.

Therefore, the domain is0,103.

04

Part (c) Step 1. Plot the function.

On plotting the function, we get,

From the graph, we can say that A is smallest atx鈮�2.08.

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Most popular questions from this chapter

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.

f(x)=-x

Exploration Graph y=x. Then on the same screen graph y=-x. What pattern do you observe? Now try y=2x+1 and y=2(-x)+1. What do you conclude?

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Determine whether each relation represents a function. For each function, state the domain and range.

4,-1,2,1,4,2
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