ANSWER:
24 miles
STEP-BY-STEP EXPLANATION:
The first thing is to make a graph of the situation, like this:
Therefore, the function of total cost will be:
[tex]C(x)=143000\cdot\sqrt[]{x^2+12^2}+55000\cdot(29-x)[/tex]
To minimize we must calculate the derivative of the function, like this:
[tex]\begin{gathered} C^{\prime}(x)=\frac{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}[/tex]
Now we set the derivative equal to 0 and solve for d, like this:
[tex]\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=\frac{435600}{17424} \\ x=\sqrt[]{25} \\ x=5 \end{gathered}[/tex]
Distance from terminal the two types of pipe meet is 29 - x, therefore would be:
[tex]\begin{gathered} d=29-5 \\ d=24\text{ miles} \end{gathered}[/tex]
