Question

determine how many miles from the terminal the two types of pipe should meet so that the total cost is minimized

Answer 1

24 miles

Explanation:

The first thing is to make a graph of the situation, like this:

Therefore, the function of total cost will be:

`C(x)=143000\cdot\sqrt[]{x^2+12^2}+55000\cdot(29-x)`

To minimize we must calculate the derivative of the function, like this:

`\begin{gathered} C^(\prime)(x)=(d)/(dx)143000\cdot\sqrt[]{x^2+12^2}+55000\cdot(29-x) \\ C^(\prime)(x)=\frac{143000d}{\sqrt[]{x^2+144}}-55000 \end{gathered}`

Now we set the derivative equal to 0 and solve for d, like this:

`\begin{gathered} \frac{143000d}{\sqrt[]{x^2+144}}=55000 \\ 143000x=55000\sqrt[]{x^2+144} \\ 143^2x^2=55^2\cdot(x^2+144) \\ 20449x^2=3025x^2+435600 \\ 20449x^2-3025x^2=435600 \\ 17424x^2=435600 \\ x^2=(435600)/(17424) \\ x=\sqrt[]{25} \\ x=5 \end{gathered}`

Distance from terminal the two types of pipe meet is 29 - x, therefore would be:

`\begin{gathered} d=29-5 \\ d=24\text{ miles} \end{gathered}`