Q. 91 - Precalculus Enhanced with Graphing Utilities

Question

The monthly revenue R achieved by selling

x wristwatches is figured to be $R\left(x\right) = 75x - 0.2x^2$ 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 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.

Step-by-Step Solution

Answer

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

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 

Step 1: Given information

The revenue equation is $$\begin{gathered} R\left(x\right)=75x-0.2x^2 \\ C\left(X\right)=32x+1750 \end{gathered}$$

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

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

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

$$\begin{gathered} P\left(x\right)=R\left(x\right)-C\left(x\right) \\ P\left(x\right)=75x-0.2x^2-32x-1750 \\ P\left(x\right)=-0.2x^2+43x-1750 \end{gathered}$$

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

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

The corresponding profit can be given as

$$\begin{gathered} f\left(x\right)=-0.2x^2+43x-1750 \\ f(107.5)=-0.2{(107.5)}^2+43(107.5)-1750 \\ f(107.5)=561.25 \end{gathered}$$

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 $P\left(x\right)=-0.2x^2+43x-1750$

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