Calculus help?

Question

1. a lighthouse is located 8 miles off a straight coast from a poiont P. a town is located 18 miles down the seacoast from P. supplies are to be moved from the town to the lighthouse on a regular basis using a minimum amount of time . if the supplies can be moved at teh rate of 7 miles per hour on water and 25 miles per hour on land, how far from the town should a dock be constructed for shipment of supplies?

Answer 1

I would help you, but the wording's kinda throwing me off.

If you could draw a diagram, then I'll help you.

Answer 2

Let

L = speed on land = 25 miles/hour
W = speed on water = 7 miles/hour
T = total time traveled
z = distance traveled over water
18 - x = distance traveled over land

Solve for 18 - x.

We have:

z² = 8² + x² = x² + 64
z = √(x² + 64)

T = z/7 + (18 - x)/25 = √(x² + 64) + (18 - x)/25

Take the derivative to find critical values.

dT/dx = (1/7)(1/2)(2x)/√(x² + 64) - 1/25 = 0
x/{7√(x² + 64)} = 1/25
25x = 7√(x² + 64)
625x² = 49(x² + 64) = 49x² + 3136
576x² = 3136
x² = 3136/576
x = √(3136/576) = 56/24 = 7/3

So

18 - x = 18 - 7/3 = 47/3 = 15 2/3

The dock should be constructed 15 2/3 miles from the town to minimize travel time.