The cost of laying power line underwater is 3 times that of underground. an island is 4km from the shore and a power station is at a distance of 8km from the point on the shore which is closest to the island. how should the power line be laid so that the cost is a minimum?

The pipeline should come ashore √2 km toward the power station from the point on shore closest to the island.

Step-by-step explanation:

Let d represent the distance from the power station to the point where the power line comes ashore. The relative cost of the line will be ...

c = 3√(4² +(8-d)²) +d

c = 3√(80 -16d +d²) +d

Differentiating c with respect to d and setting that to zero gives ...

dc/dd = 0 = (3/2)(2d-16)/√(80 -16d +d²) +1

Subtracting 1, multiplying by the denominator, and squaring both sides gives a quadratic that can be solved in the usual way. It reduces to ...

(d -8)² -2 = 0

d = 8±√2 . . . . . . . . only the root 8-√2 is useful

This tells us the pipeline should come ashore √2 km toward the power station from the point closest to the island.

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).