Question

a small resort is situated on an island that lies exactly 4 miles from , the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from is the closest source of fresh water. we want to efficiently transport this fresh water to the resort. if it costs 1.7 times as much money to lay pipe in the water as it does on land, how far down the shoreline from should the pipe from the island reach land in order to minimize the total construction costs? distance from

Answer 1

Minimize the total construction costs, the pipe from the island should reach land at a distance of approximately 9.17 miles down the shoreline from .

Step-by-step explanation:

To minimize the total construction costs, the pipe from the island should reach land at a distance that balances the cost of laying the pipe on land and in the water. Let's assume that the cost of laying 1 mile of pipe on land is $x. Therefore, the cost of laying 1 mile of pipe in the water would be $1.7x.

By using the Pythagorean theorem, we can determine the distance from the island to the point where the pipe reaches land. The distance along the shoreline is 10 miles, and the distance from the island to the shoreline is 4 miles. Therefore, using the Pythagorean theorem, we have:

d = sqrt((10)^2 - (4)^2)

Simplifying:

d = sqrt(100 - 16)

d = sqrt(84)

d ≈ 9.17 miles

Therefore, the pipe from the island should reach land at a distance of approximately 9.17 miles down the shoreline from .