Question

A luxury hotel group purchases a deserted island 3. 5 miles offshore and due south of an endangered bird nesting beach. The nearest power source is on land, 10 miles due east of the bird nesting beach. Surveying for electricity construction costs $250 per mile over the water and $150 per mile on the ground. As a surveyor, the cheapest price (in dollars) for which you can bill the job involves surveying above the water to a point inbetween the nesting beach and the power source, and then from the nesting beach to the power source. Find the least expensive price for which you can bill the job

Answer 1

Answer:

T.C ( 1.1007 ) = T.C_min = $2252  

Step-by-step explanation:

Given:

- Cost of electricity construction over water C_w = $250 / mile

- Cost of electricity construction over ground C_w = $150 / mile

- The distance from hotel island = 3.5 miles

- The distance from beach to power source = 10 miles

Find:

Find the least expensive price for which you can bill the job

Solution:

- This problem requires cost optimization. So we need to develop a cost function as follows

Total Cost = C_w*(distance from island to x) + C_g*( distance from x to power)

- Now calculate the relevant distances using Pythagoras theorem:

              Distance from island to x = sqrt ( x^2 + 3.5^2 )

              Distance from x to power station = 10 - x

- Input the distances in the cost function:

              T.C ( x ) = C_w*sqrt ( x^2 + 3.5^2 ) + C_g*(10 - x )

- Input the relevant rates:

              T.C ( x ) = 250*sqrt ( x^2 + 3.5^2 ) + 150*(10 - x )  

- Simplify:

              T.C ( x ) = 250*sqrt ( x^2 + 12.25 ) + 1500 - 150x

- Next, we will optimize the cost to minimum. We need the distance x that would give us the minimum cost. To minimize the function, set its derivative with respect to x equals to zero.

                T.C' ( x ) = 500*x*( x^2 + 12.25 )^( -0.5 ) - 150

- Set the derivative to zero and solve for x:

                            sqrt ( x^2 + 12.25 ) = 10x/3

Squaring both sides:

                           9*x^2 + 110.25 = 100*x^2

Simplify and solve:

                      x = sqrt (110.25 / 91) = 1.1007 miles

- The cost function is minimized at x = 1.1007 miles. We will input this back into our function and evaluate the minimum cost as follows:

                 T.C ( 1.1007 ) = 250*sqrt ( 1.1007^2 + 12.25 ) + 1500 - 150*1.1007

                 T.C ( 1.1007 ) = T.C_min = $2252  

                                 

- So the minimum cost associated with this plan is $2252.