Can someone solve this problem for me please

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 8000 dollars per mile to lay the pipe under the water and 5000 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  that will minimize the total cost.

x=

Answer 1

Let x = the distance from point B east to point Q.

The distance form the island house, H, to point Q is the hypotenuse of the right triangle HBQ.

d_HQ = √(x² + 6²)

The distance from point Q to the the Water, W, is:

d_QW = 8-x

C(x) = 8000√(x² + 36) + 5000(8-x)

To find the minimum cost, take the derivative of C(x), set that to zero and solve for x.

C'(x) = 8000(2x)/(2√(x² + 36)) - 5000x

8000x/√(x² + 36) - 5000 = 0

8000x = 5000√(x² + 36)

8x/5 = √(x² + 36)

Squaring both sides

64x²/25 = x²+36

39x²/25 = 36

x = 6√(25/39) = 30√(1/39)≈4.8 miles east of point B

Note: C(4.8) ≈ $77470

Answer 2

Cost function C(t) = 8000(36 + x²)^1/2 + 5000(8 - x).

To get minimum let differentiate this function: C'(x) = 8000x/(36 + x²)^1/2 - 5000 = 0; 8000x = 5000(36 + x²)^1/2;

8x = 5(36 + x²)^1/2; 64x² = 25(36 + x²); 39x² = 900; x² = 300/13; x = ± 4.8.

To prove that for x = 4.8 miles we have minimum C''(x) = 8000(36 + x²)^1/2 - 4000x²(36 + x²)^-3/2=

4000(72 + x²)/(36 + x²)^3/2.

Clear that C(4.8) > 0, so we have minimum cost if x = 4.8 miles