Q. 87 - Precalculus Enhanced with Graphing Utilities

Question

Suppose that the manufacturer of a

gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is $R\left(p\right)=-4p^2 + 4000p$

What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?

Step-by-Step Solution

Answer

The unit price should be $500 and the maximum revenue is R(x)=1000000

Step 1: Given information

We are given a equation $R\left(p\right)=-4p^2 + 4000p$

Step 2: Find the vertex

The coefficient of the square term is negative hence the parabola opens downwards hence the maximum value is at the vertex

The vertex can be given as

$$\begin{gathered} x=-\frac{b}{2a} \\ x=\frac{4000}{8} \\ x=500 \end{gathered}$$

Hence the price of each unit should be 

$500

Step 3: To find the revenue

substitute 500 for  p in the equation

$$\begin{gathered} R\left(p\right)=-4{\left(500\right)}^2+4000\left(500\right) \\ R\left(p\right)=-100000+200000 \\ R\left(p\right)=1000000 \end{gathered}$$

The revenue is $$1000000$

Step 4: Conclusion

The price of a single unit is $500

And the total revenue is $1000000