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Q. 91

Expert-verified
Found in: Page 158

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

The monthly revenue R achieved by sellingx wristwatches is figured to be $R\left(x\right)=75x-0.2{x}^{2}$ The monthly cost C of selling x wristwatches is$C\left(x\right)=32x+1750$(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 maximizeprofit? 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 $P\left(x\right)=-0.2{x}^{2}+43x-1750$ 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

See the step by step solution

Step 1: Given information

The revenue equation is $R\left(x\right)=75x-0.2{x}^{2}\phantom{\rule{0ex}{0ex}}C\left(X\right)=32x+1750$

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

$x=-\frac{b}{2a}\phantom{\rule{0ex}{0ex}}x=\frac{75}{0.4}\phantom{\rule{0ex}{0ex}}x=187.5$

The corresponding revenue is

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

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

We get

$P\left(x\right)=R\left(x\right)-C\left(x\right)\phantom{\rule{0ex}{0ex}}P\left(x\right)=75x-0.2{x}^{2}-32x-1750\phantom{\rule{0ex}{0ex}}P\left(x\right)=-0.2{x}^{2}+43x-1750$

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

$x=-\frac{b}{2a}\phantom{\rule{0ex}{0ex}}x=\frac{43}{0.4}\phantom{\rule{0ex}{0ex}}x=107.5$

The corresponding profit can be given as

$f\left(x\right)=-0.2{x}^{2}+43x-1750\phantom{\rule{0ex}{0ex}}f\left(107.5\right)=-0.2{\left(107.5\right)}^{2}+43\left(107.5\right)-1750\phantom{\rule{0ex}{0ex}}f\left(107.5\right)=561.25$

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 $P\left(x\right)=-0.2{x}^{2}+43x-1750$

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