Hard 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 7 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 5000 dollars per mile to lay the pipe under the water and 3125 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

Step 1: Define Variables

1. Let x be the distance in miles from point B to point Q on the shoreline.
2. The total distance from point B to the water source is 8 miles, so the distance from Q to the water source is 8−x miles.

Step 2: Express the Distances

1. The underwater distance from the island to Q can be calculated using the Pythagorean theorem. The distance from the island to B is 7 miles, so the underwater distance from the island to Q is: x^2+7^2=x^2+49 miles.√(x^2 + 7^2)=√( x^2 + 49) miles.

Step 3: Express the Total Cost

1. The cost of laying the pipe underwater is $5000 per mile.
2. The cost of laying the pipe on land is $3125 per mile.
3. The total cost C(x) as a function of x is: C(x)=5000⋅√(x^2+49)+3125⋅(8−x)

Step 4: Differentiate the Cost Function

To find the value of x that minimizes the cost, we take the derivative of C(x) with respect to x and set it equal to zero:

C′(x)=5000⋅x/(x^2+49)−3125.

Set C′(x)=0 to find the critical point:

5000⋅x/(x^2+49)−3125=0

x/(x^2+49) =3125/5000

x/(x^2+49) =5/8

Step 5: Solve for x

Square both sides to eliminate the square root:

x^2/(x^2+49)=25/64

Solve this equation. I found .x≈5.6 miles.

Step 6: Interpretation

The value of x that minimizes the total construction cost is approximately 5.6 miles east of point B,

This means that point Q should be located approximately 5.6 miles east of point B to minimize the total construction cost.

Answer 2

≤Although you've listed this as a "hard" problem, it's a fairly standard exercise from first-semester Calculus.

As I mentioned above, Ross M's solution is incomplete, even though it happens to give the correct answer in this case. Just because you only get one interior critical point (x = 5.6) does not mean it's the extremum that you're looking for!!! If the problem had asked for the MAXIMUM cost (perhaps you're consulting for a construction supply company and want to sell a lot of materials?), this would occur at x = 0 (assuming Mark's computations are correct).

But as far as finding the minimum, if you follow Ross' method in general, the critical point with x > 0 is given by the formula:

x = D / sqrt( (U/G)^2 - 1 )

where D is the distance to the island, U is the cost per distance underwater, and G is the cost per distance underground. If U ≤ G, there is no such critical point (which makes sense, as there's no cost benefit to building underground in this case). It's also possible that there is a critical point, but that it's larger than the distance along the shore (this happens when the underwater cost is extremely high relative to the underground cost, and the minimum occurs at x = 0).

For a fun variant on this problem:

https://web.williams.edu/Mathematics/sjmiller/public_html/103/Pennings_DogsCalculus.pdf