Q. 67 - Precalculus Enhanced with Graphing Utilities

Question

The number $N$ of cars produced at a certain factory in one day after $t$ hours of operation is given by $N\left(t\right)=100t-5t^2, 0\le t\le 10.$ If the cost $C$ (in dollars) of producing N cars is $C\left(N\right)=15,000+8000N,$ find the cost $C$ as a function of the time $t$ of operation of the factory.

Step-by-Step Solution

Answer

Hence, $C\left(t\right)=15,000+8,00,000t-40,000t^2.$

Step 1. Given Information

Given that $N\left(t\right)=100t-5t^2, 0\le t\le 10$ and $C\left(N\right)=15,000+8000N.$

Step 2. Solution

We have to find the cost $C$ as a function of time $t$ that is we have to find $C\circ N\left(t\right).$

Here, $C\left(N\right)=15,000+8000N$ and $N\left(t\right)=100t-5t^2.$

So,

$$\begin{gathered} C\circ N\left(t\right)=C\left(N\right(t\left)\right) \\ C\circ N\left(t\right)=15,000+8000\left(N\right(t\left)\right) \\ C\circ N\left(t\right)=15000+8000\{100t-5t^2\} \\ C\circ N\left(t\right)=15,000+8,00,000t-40,000t^2 \\ C\left(t\right)=15,000+8,00,000t-40,000t^2. \end{gathered}$$

Step 3. Final Answer

Hence, C as a function of time $t$ is: -

$C\left(t\right)=15,000+8,00,000t-40,000t^2.$