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 1.4 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?
P=point on shore
R=resort
W=water source
X=point on shore where pipe enters water
Assume land pipe costs 1, water pipe costs 1.4
If X is at distance x from P, then
the cost of the pipeline
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
dc/dx = 0 where
sqrt(x^2+36) = 1.4x
x^2+36 = 1.96x^2
.96x^2 = 36
x^2 = 37.5
x = 6.12
Well, that's quite a pipe dilemma! Let me think... If the resort wants to minimize construction costs, they should aim to balance the cost of laying pipe on land with laying pipe in water.
Since it costs 1.4 times as much money to lay pipe in water, we can assume that laying pipe in water is the more expensive option. So, the resort should try to minimize the distance of the pipe in water.
Now, the key here is to find the perfect sweet spot along the shoreline. If the pipe reaches land too close to the island, they would have to lay more pipe on land, which would be cheaper. If the pipe reaches land too close to the source of fresh water, they would have to lay more pipe in water, which would be more expensive.
Let's do some calculations. The total cost of pipe construction can be represented by the equation:
Cost = Cost of pipe on land + Cost of pipe in water
Since the cost of pipe in water is 1.4 times the cost of pipe on land, we can write:
Cost of pipe in water = 1.4 * Cost of pipe on land
Now, let's represent the distance from P where the pipe reaches land as x. The distance from P to the source of fresh water is given as 10 miles. So, the distance of pipe in water would be 6 miles - x (since it is exactly 6 miles from P to the island).
Finally, we can substitute these values into our equations:
Cost = Cost of pipe on land + Cost of pipe in water
Cost = Cost of pipe on land + 1.4 * Cost of pipe on land
Cost = 2.4 * Cost of pipe on land
So, we want to minimize the total cost, which means we need to minimize the Cost of pipe on land. This would happen when the distance from P where the pipe reaches land, x, is maximized.
Therefore, the pipe from the island should reach land as close to the source of fresh water as possible, 10 miles down the shoreline from P. This way, there would be no cost for laying pipe in water, and the cost would be minimized.
Voila! The resort can save some serious construction costs by keeping the pipe out of the water. Just remember, in the piping world, dry land is always cheaper than diving into the deep blue sea!
To minimize the total construction costs, we need to find the optimal point on the shoreline where the pipe from the island should reach land. Let's break down the problem and figure out the steps to solve it.
1. Let's assume that the point where the pipe reaches the shoreline is X miles down the shoreline from P. We need to find the value of X that minimizes the construction costs.
2. The cost of laying pipe on land is proportional to the distance on land. Therefore, the cost of laying pipe on land is 1 times X or simply X.
3. The cost of laying pipe in water is 1.4 times the distance in water. The distance in water is given by the distance from the island to the point X. Since the island is 6 miles away from the nearest point on the shoreline, the distance in water is X + 6 miles.
4. Therefore, the cost of laying pipe in water is 1.4 times (X + 6) or 1.4X + 8.4.
5. The total construction cost is the sum of the cost of laying pipe on land and the cost of laying pipe in water. So, the total cost function is given by:
Total cost = Cost on land + Cost in water
= X + (1.4X + 8.4)
= 2.4X + 8.4
6. To find the value of X that minimizes the total construction cost, we need to find the derivative of the total cost function with respect to X and set it equal to zero. Then solve for X.
d(Total cost)/dX = 2.4 = 0
2.4X + 8.4 = 0
2.4X = -8.4
X = -8.4 / 2.4
X ≈ -3.5
7. Since X represents the distance down the shoreline, it cannot be negative. Therefore, we ignore the negative value and choose the closest positive distance to minimize the cost.
Therefore, the optimal distance down the shoreline from P where the pipe from the island should reach land to minimize the total construction costs is approximately 3.5 miles.
Note: It's important to consider any additional factors or constraints that may affect the decision-making process when solving real-world problems. This explanation focuses on the mathematical approach to find the solution.
That would be because they missed the 2 in the denominator for their:
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
it should actually be:
dc/dx = -1 + 1.4/2*sqrt(x^2+36)