Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

A wire $10$ meters 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 A enclosed 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 $A = A (x)$ . For what value of x is A smallest?

Part (a): Total area A enclosed by the pieces of wire as a function is $A (x) = x^{2} + \frac{25 - 20 x + 4 x^{2}}{π}$ .

Part (b): The domain of A is $(0, \frac{5}{2})$ .

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

The value of x for which A is smallest at $x \approx 1.4$ m.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1. Given information.

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

Perimeter $(p) = 4 x$

From the given figure,

Circumference of the circle formed is $10 - 4 x$ m.

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

localid="1645810318472" $2 πr (x) = 10 - 4 x r (x) = \frac{10 - 4 x}{2 π} r (x) = \frac{5 - 2 x}{π}$

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

Consider the given question,

$A (x) = x^{2} + {πr}^{2} A (x) = x^{2} + π {(\frac{5 - 2 x}{π})}^{2} A (x) = x^{2} + \frac{25 - 20 x + 4 x^{2}}{π}$

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

Consider the given question,

$r (x) = \frac{5 - 2 x}{π}$ . Also $r (x) > 0$ . Then,

$\frac{5 - 2 x}{π} > 0 5 - 2 x > 0 x < \frac{5}{2}$

x cannot be negative. As length cannot be negative.

Therefore, the domain is $(0, \frac{5}{2})$ .

Part (c) Step 1. Plot the function.

On plotting the function, we get,

From the graph, we can say that A is smallest at $x \approx 1.4$ m.

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Step by Step Solution

Part (a) Step 1. Given information.

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

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

Part (c) Step 1. Plot the function.

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Step by Step Solution

Part (a) Step 1. Given information.

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

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

Part (c) Step 1. Plot the function.

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