A small resort is situated on an island that lies exactly 5 miles from P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest source of fresh water. If it costs 1.5 times as much money to lay pipe in the water as it does on land, how far down the shoreline from P should the pipe from the island reach land in order to minimize the total construction costs?
If the distance is x from P, then the cost is
c(x) = 1.5√(x^2+25) + (10-x)
dc/dx = 1.5x/√(x^2+25) - 1
= [1.5x - √(x^2+25)]/√(x^2+25)
dc/dx=0 when the numerator is zero, or
1.5x = √(x^2+25)
2.25x^2 = x^2+25
1.25x^2 = 25
x^2 = 20
x = 2√5
To minimize the total construction costs, we need to find the point on the shoreline where the sum of the costs for laying the pipe on land and in the water is the lowest.
Let's denote the distance from P to the point where the pipe reaches land as x miles.
The distance between the island and the starting point of the pipe on land is (5 - x) miles because the island is 5 miles from P.
The cost of laying the pipe on land is x miles * cost per mile on land.
The cost of laying the pipe in the water is (5 - x) miles * cost per mile in the water.
Given that the cost of laying the pipe in the water is 1.5 times the cost of laying it on land, the cost per mile in the water is 1.5 * cost per mile on land.
Hence, the total cost function can be expressed as:
Total cost = x * (cost per mile on land) + (5 - x) * 1.5 * (cost per mile on land)
To find the distance x that minimizes the total cost, we need to minimize this cost function.
Let's simplify the cost function:
Total cost = x * (cost per mile on land) + (7.5 - 1.5x) * (cost per mile on land)
= (x + 7.5 - 1.5x) * (cost per mile on land)
= (7.5 - 0.5x) * (cost per mile on land)
Since we are only interested in finding the distance x, we can drop the constant factor (cost per mile on land) because it doesn't affect the value of x that minimizes the total cost.
So, the simplified cost function is:
Total cost = 7.5 - 0.5x
To minimize the total cost, we need to find the value of x that minimizes this function. The minimum occurs at the peak of the curve, which in this case will be at the midpoint of the interval (0, 7.5).
The midpoint of the interval is (0 + 7.5) / 2 = 3.75.
Therefore, the pipe from the island should reach land approximately 3.75 miles down the shoreline from point P in order to minimize the total construction costs.