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 6000 dollars per mile to lay the pipe under the water and 3750 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

Answer 1

Answer:

Q should be 4.8 miles east of B.

Explanation:

As the diagram shows, we can simply express the construction cost as a function of angle θ (represented in the diagram).  

The length of the pipe in water (represented with blue) = 6/cos θ

The length of the pipe on land (represented with brown) = (8-6*tan θ)

Construction Cost = (6/cos θ) (6000) + (8-6*tan θ)(3750)

The above function represents the construction cost and is in terms of θ

The value of θ varies from 0 degrees to 53.13 degrees as per the diagram.

We can take the derivative of Construction Cost function with respect to θ and equate it to zero to find the angle θ at which the construction cost is minimum.

d(Construction Cost)/d θ = -4500(5*sec θ – 8*tan θ)(sec θ)

-4500(5*sec θ – 8*tan θ) (sec θ) = 0

θ = 38.68 degrees

Using the value of θ, we can find the distance of Q from B.

Distance of Q from B = 6*tan θ  

Distance of Q from B = 6*tan (38.68 degrees)

Distance of Q from B = 4.8 miles