need help on optimization problem?

Question

you need to run a cable underground from point A to poing B at the edge of a park that is situated on the city block corner. Point A is 150 feet from the corner and B is 120ft from the corner. it is cheaper to dig along the street than through the park, but the city will not allow any digging on first Ave. it cost $30 per ft to run the cable along Main street and $65 per ft to run it through the park

Determine how far the cable should be run along main street before cutting through the park to minimize the cost. show all work. you can use your calc as you wish, but clearly explain what you graphed and where you looked to get your answer.

Answer 1

d = x + √((150 - x)^2 + 120^2)
C = 30x + 65√((150 - x)^2 + 120^2)
C = 30x + 65(22500 - 300x + x^2 + 14400)^(1/2)
C = 30x + 65(36900 - 300x + x^2)^(1/2)
dC/dx = 30 + (1/2)(65)(2x - 300) (36900 - 300x + x^2)^-(1/2)
65(2x - 300)/2 = - 30(36900 - 300x + x^2)^(1/2)
130^2(x - 150)^2 = 3600(36900 - 300x + x^2)
130^2(x^2 - 300x + 22500) = 3600(36900 - 300x + x^2)
169(x^2 - 300x + 22500) = 36(36900 - 300x + x^2)
169x^2 - 50700x + 3802500 = 1328400 - 10800x + 36x^2
133x^2 - 39900x + 2474100 = 0
x^2 - 300x + 18602.26 = 0
x = (300 - √(90000 - 74409.023))/2
x = (300 - √15590.977443609)/2
x = (175.1361)/2
x = 87.568 ft.

87.56 ft. $11,419.53757
87.57 f t. $11,419.53756
87.58 ft. $11,419.53758

Answer 2

You haven't given enough information to allow this problem to be solved. Is Point A on First Ave? Is Point B on Main Street? Or vice versa? Do First Ave and Main Street intersect at right angles?

Let D be the distance that the cable is run along Main St. I will assume that point A is on Main and point B is on First Ave, and that the streets do indeed intersect at right angles. I have also assumed that you can follow any route you like through the park.

The total cost of laying the cable, which is the function you are trying to minimise, will be $30 x D + $65 x (sqrt( (150 - D)^2 + 120^2)). Given the way the question is worded, I would suggest you trying graphing that function over the range D = [0,150] and find the minimum. There is probably some arithmetic that could be used to solve for the exact value, but I am too lazy to figure that out for you.

Answer 3

Your question is a bit unclear, but I think I figured out what you mean:

Let's say that x is the length of cable run along Main Street.
Then use Pythagorus to determine length run through park:
sqrt(120^2 + (150 - x)^2) = sqrt(14400 + 22500 - 300x + x^2) = sqrt(x^2 - 300x + 36900)

Total cost: 30x + 65*sqrt(x^2 - 300x + 36900)

y' = 30 + 32.5*(2x - 300)/sqrt(x^2 - 300x + 36900) = 0 (max)

sqrt(x^2 - 300x + 36900)*30 + 65x - 9750 = 0
6*sqrt(x^2 - 300x + 36900) = 1950 - 13x
36*(x^2 - 300x + 36900) = 3802500 - 50700x + 169x^2
133x^2 - 39900x + 2474100 = 0

x = 87.568