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.
