Please help with this word problem

Question

A small house was built on an island off a perfectly straight shoreline. The point B on the shoreline that is closest to the island is exactly 6 miles from the island. Eight miles east of B is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between B and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 7000 dollars per mile to lay the pipe under the water and 4375 dollars per mile to lay pipe on land. How far east of B should Q be located in order to minimize the total construction costs?

Find a formula for the cost of laying the pipe in terms of 𝑥

𝐶(𝑥) =   

Find the value for 𝑥 that will minimize the total cost.

𝑥 =  

Answer 1

This question falls under optimization and requires a few steps.

Step 1) Create a function that represents cost as a function of x. There are 2 distances in this question, in the water and out of the water. The distance in the water can be found with the familiar pythagorean theorem, (6²+x²)^1/2 will be that distance. The distance on land is simply (8-x).

With this in mind, the Cost function is C(x) = 7000*(6²+x²)^1/2 + 4375*(8-x).

Step 2) As I mentioned earlier, this is an optimization problem. Therefore, it requires finding a derivative, so you will find the derivative with respect to x.

C'(x) = 7000*x / (6²+x²)^1/2 - 4375.

Step 3) As this is an optimization question, we are looking for where the derivative is equal to 0. As a reminder, a derivative equaling 0 means you have reached a point of 0 slope, typically a minimum or a maximum (in this case, a minimum). So the next step is as follows:

0 = 7000*x / (6²+x²)^1/2 - 4375

For this part, use either a graphing calculator and the intercept function to find your zero OR use Desmos (which I highy recommend for homeworks. However, this wouldn't be available for an exam).

This leaves you with x = 4.803 miles.

*Step 4) To verify this result, you can graph the initial cost function and look at where the minimum is.