Q. 70 - Precalculus Enhanced with Graphing Utilities

Question

The price $p$, in dollars, of a certain commodity and the quantity $x$ sold obey the demand equation $p=-\frac{1}{5}x+200, 0\le x\le 1000.$ Suppose that the cost C, in dollars, of producing $x$ units is $C=\frac{\sqrt{x}}{10}+400$.

Assuming that all items produced are sold, find the cost C as a function of the price $p$. 

Step-by-Step Solution

Answer

The cost function, $C\left(p\right)=\frac{\sqrt{-5p+1000}}{10}+400.$

Step 1. Given Information

Given that $p\left(x\right)=-\frac{1}{5}x+200, 0\le x\le 1000,$ and $C\left(x\right)=\frac{\sqrt{x}}{10}+400.$

Step 2. Solution

Here, $p\left(x\right)=-\frac{1}{5}x+200, 0\le x\le 1000$ and $C\left(x\right)=\frac{\sqrt{x}}{10}+400.$

We have to find the cost $C$ as a function of $p.$

Now,

$$\begin{gathered} p=-\frac{1}{5}x+200, 0\le x\le 1000 \\ p-200=-\frac{1}{5}x \\ 5(p-200)=-x \\ x\left(p\right)=-5p+1000, 0\le p\le 200. \end{gathered}$$

C as a function of price $p$ is $C\circ x\left(p\right).$ 

So,

$$\begin{gathered} C\circ x\left(p\right)=C\left(x\right(p\left)\right) \\ C\circ x\left(p\right)=\frac{\sqrt{x\left(p\right)}}{10}+400 \\ C\circ x\left(p\right)=\frac{\sqrt{-5p+1000}}{10}+400 \\ C\left(p\right)=\frac{\sqrt{-5p+1000}}{10}+400 \end{gathered}$$

Step 3. Final answer

Hence, $C\left(p\right)=\frac{\sqrt{-5p+1000}}{10}+400.$