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. 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 should the pipe from the island reach land in order to minimize the total construction costs?
Draw a diagram, where
R is the resort
P is the named point on shore
W is the water well
L is the point on land where the pipe comes ashore
RPL is a right triangle, where
RP=4
PL = x
PW=10, so LW=10-x
without loss of generality, we may assume the land cost is 1, so the total cost is
c = 1.5√(16+x^2) + (10-x)
dc/dx = 1.5x/√(16+x^2) - 1
dc/dx = 0 when x = 8/√5 = 3.58
To minimize the total construction costs, we need to find the point along the shoreline that minimizes the sum of the costs of laying pipe on land and in the water.
Let's assume that the pipe reaches land x miles down the shoreline from point P. We can now use the concept of similar triangles to set up a ratio between the length of pipe in the water and on land.
The ratio of the length of pipe in the water to the length of pipe on land can be represented by the ratio of the distances from the island to the pipe's landing point and from the landing point to the source of fresh water.
Let's represent the distance from the island to the landing point as d1 (in miles) and the distance from the landing point to the source of fresh water as d2 (in miles). Since we know that the distance from the island to the landing point is x miles, we have:
d1 = x
Also, we know that the distance from the landing point to the source of fresh water is 10 miles, so:
d2 = 10 - x
Now, let's set up the ratio:
Length of pipe in water : Length of pipe on land = d1 : d2
Length of pipe in water : Length of pipe on land = x : (10 - x)
Since we are given that it costs 1.5 times as much money to lay pipe in the water as it does on land, we can set up the cost ratio:
Cost ratio = (Cost of pipe in water) : (Cost of pipe on land) = 1.5 : 1
To minimize the total construction costs, the length ratio should be equal to the cost ratio:
Length of pipe in water : Length of pipe on land = 1.5 : 1
x : (10 - x) = 1.5 : 1
Next, let's cross multiply:
1.5x = 1(10 - x)
1.5x = 10 - x
Adding x to both sides:
2.5x = 10
Dividing both sides by 2.5:
x = 4
Therefore, the pipe should reach land 4 miles down the shoreline from point P in order to minimize the total construction costs.
To minimize the total construction costs, we need to find the point on the shoreline that minimizes the sum of the pipe costs on land and in water.
Let's break down the problem step by step:
1. We know that the distance from the island to the nearest point on the shoreline is 4 miles. This segment between the island and the shoreline is the same as the distance our pipe will have to cross in the water.
2. The distance between the nearest point to the island along the shoreline and the closest source of fresh water is 10 miles. This is the distance our pipe will have to run on land.
3. The cost of laying pipe on land is the same as the distance on land multiplied by the cost factor. Let's assume the cost factor for laying pipe on land is X, so the cost of laying pipe on land is 10 * X.
4. The cost of laying pipe in the water is 1.5 times the cost of laying pipe on land. So the cost of laying pipe in the water is 1.5 * (4 * X) = 6 * X.
Now, we can express the total construction cost as the sum of the cost on land and in the water:
Total Cost = Cost on Land + Cost in Water
Total Cost = 10 * X + 6 * X
Simplifying the equation, we get:
Total Cost = 16 * X
To minimize the total construction cost, we need to minimize X in the equation.
Since X represents the cost factor for laying pipe on land, we have to minimize the distance on land.
Given that the distance from the island to the nearest point on the shoreline is 4 miles, we need to determine how far down the shoreline from the nearest point to should the pipe reach land.
This can be calculated by subtracting 4 miles from the total distance between the nearest point to the freshwater source:
Distance on Land = (10 - 4) miles
Distance on Land = 6 miles
Therefore, the pipe from the island should reach land 6 miles down the shoreline from the nearest point to minimize the total construction costs.