A small resort is situated on an island that lies exactly 6 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 2.3 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?
To minimize the total construction costs, we need to find the point on the shoreline where the pipe from the island should reach land. Let's call this point Q.
Let's split the problem into two cases:
Case 1: Pipe reaching land before the source of fresh water:
In this case, the distance from P to Q is less than the distance from Q to the source of fresh water. Let's call the distance from P to Q as x.
The distance from Q to the source of fresh water would then be (10 - x).
Case 2: Pipe reaching land after the source of fresh water:
In this case, the distance from P to the source of fresh water would be less than the distance from P to Q. Let's call the distance from P to the source of fresh water as y.
The distance from Q to the source of fresh water would then be (10 + y).
Now, let's calculate the cost for each case:
Case 1: Cost of pipe in water = 2.3 * x.
Cost of pipe on land = x (since pipe is laid on land from P to Q).
Total cost for case 1 = 2.3x + x = 3.3x.
Case 2: Cost of pipe in water = 2.3 * y.
Cost of pipe on land = y (since pipe is laid on land from P to the source of fresh water).
Total cost for case 2 = 2.3y + y = 3.3y.
To minimize the total construction cost, we need to compare the costs for both cases and choose the one with the lower cost.
For case 1, the total cost is 3.3x, and for case 2, the total cost is 3.3y.
Since x < y (because Q is closer to P than the source of fresh water), we can deduce that the cost for case 1 will be less than the cost for case 2.
Therefore, the pipe from the island should reach land before the source of fresh water.
To find the optimal distance x, we can set up an equation:
3.3x = 2.3(10 - x)
Simplifying the equation:
3.3x = 23 - 2.3x
5.6x = 23
x ≈ 4.107 miles
So, the pipe from the island should reach land around 4.107 miles from P to minimize the total construction costs.
To find the distance down the shoreline from P where the pipe from the island should reach land in order to minimize the total construction costs, we can solve this problem using calculus.
Let's assume that the island is located at point I, and the point where the pipe reaches land is at point L. We want to find the distance LI that minimizes the total construction cost.
First, let's define some variables:
- Let x be the distance from P to L (in miles).
- The cost of laying pipe on land is $1 per mile.
- The cost of laying pipe in water is $2.3 per mile.
Now, let's set up the equation for the total construction cost, C(x):
C(x) = 1 * x + 2.3 * (6 - x)
- The first term, 1 * x, represents the cost of laying pipe on land. Since the cost is $1 per mile, we simply multiply the distance x by 1.
- The second term, 2.3 * (6 - x), represents the cost of laying pipe in water. Since the cost is $2.3 per mile, we multiply the distance in water (which is 6 - x) by 2.3.
To find the distance x that minimizes the total construction cost, we'll need to find the minimum of the function C(x) using calculus.
Differentiate C(x) with respect to x:
C'(x) = 1 - 2.3
Set C'(x) equal to zero and solve for x:
1 - 2.3 = 0
-1.3 = 0
x = 6
Since the derivative of C(x) is a constant (-1.3) and does not depend on x, this means that C(x) is a linear function. Therefore, the minimum construction cost is obtained when the pipe reaches land at a distance of x = 6 miles from P, the nearest point to the island along the shoreline.
Hence, the pipe should reach land exactly 6 miles down the shoreline from P to minimize the total construction costs.
Draw a diagram. Label the points
R = resort
P = closest point
W = water source
X = place where the pipeline comes ashore
So let x = PX
Let the cost of laying pipe on land be 1. Then
the cost of laying pipe is
c = 2.3 * √(6^2+x^2) + 10-x
dc/dx = 23/√(x^2+36) - 1
c is minimum when x = 20√(3/143)