Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

The monthly revenue R achieved by selling
x wristwatches is figured to be $R (x) = 75 x - 0.2 x^{2}$ The monthly cost C of selling x wristwatches is $C (x) = 32 x + 1750$
(a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue?
(b) Profit is given as P(x) = R(x) - C(x). What is the profit function?
(c) How many wristwatches must the firm sell to maximize
profit? What is the maximum profit?
(d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

a)To maximize the revenue the firm must sell 188 wristwatch and the maximum revenue is about $7031

b)The profit function can be given as $P (x) = - 0.2 x^{2} + 43 x - 1750$

c) They should sell 108 watches and the profit is $561

d)As the function of profit depends on the cost function and it does not depend on revenue function Hence the maximum values are different

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

The revenue equation is $R (x) = 75 x - 0.2 x^{2} C (X) = 32 x + 1750$

Step 2: Find the vertex

In the revenue equation the coefficient of $x^{2}$ term is negative. hence the parabola opens downwards. the maximum value is at the vertex

The vertex can be given as

$x = - \frac{b}{2 a} x = \frac{75}{0.4} x = 187.5$

The corresponding revenue is

$f (187.5) = 75 (187, 5) - 0.2 {(187.5)}^{2} f (187.5) = 7031.25$

Part b) Step 1: Find the profit function by substituting the corresponding function in the equation

We get

$P (x) = R (x) - C (x) P (x) = 75 x - 0.2 x^{2} - 32 x - 1750 P (x) = - 0.2 x^{2} + 43 x - 1750$

Part c) Step 1) Find the vertex of the quadratic equation of profit

As the coefficient of $x^{2}$ term is negative the parabola opens downward hence the maximum value is at the vertex

The vertex can be given as

$x = - \frac{b}{2 a} x = \frac{43}{0.4} x = 107.5$

The corresponding profit can be given as

$f (x) = - 0.2 x^{2} + 43 x - 1750 f (107.5) = - 0.2 {(107.5)}^{2} + 43 (107.5) - 1750 f (107.5) = 561.25$

Part d) Step 1: Explanation

As the function of profit depends on the cost function and it does not depend on revenue function Hence the maximum values are different

Step 6: Conclusion

a)They should sell 188 wristwatches and the revenue is around 7031

b)The profit equation can be given as $P (x) = - 0.2 x^{2} + 43 x - 1750$

c)To make the maximum profit they should sell 108 watches and the profit is $561

d)As the function of profit depends on the cost function and it does not depend on revenue function

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Step by Step Solution

Step 1: Given information

Step 2: Find the vertex

Part b) Step 1: Find the profit function by substituting the corresponding function in the equation

Part c) Step 1) Find the vertex of the quadratic equation of profit

Part d) Step 1: Explanation

Step 6: Conclusion

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Given information

Step 2: Find the vertex

Part b) Step 1: Find the profit function by substituting the corresponding function in the equation

Part c) Step 1) Find the vertex of the quadratic equation of profit

Part d) Step 1: Explanation

Step 6: Conclusion

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