You have been asked to determine where a water works should be built along a river between Chesterville and Denton to minimize the total cost of the project. The pipe to Chesterville costs $3000 per mile and the pipe to Denton costs $7000 per mile. Find the length of each pipe so that the total cost is a minimum. What is the cost?

Length of pipe to Chesterville is 8.376 miles and

Length of pipe to Denton is 5.46 miles

Step-by-step explanation:

Here we have

The distance of Chesterville from the river is 3 miles, while the distance of Denton from the river is 5 miles

The bank of the river is 10 miles long

Therefore, we have

If x is the distance from the point directly opposite to Chesterville to the location of the water works, the equation is;

Cost to Chesterville =

Cost to Denton =

Total cost is then;

We differentiate the above equation and equate it to zero to get the minimum cost as

= 0

Solving the quartic equation we get

x = 7.82 miles

Therefore the length of is given as

Length of pipe to Chesterville  

Length of pipe to Denton = .

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