Calculus 1 Optimization Problem

[SittsK]

A utility company needs to hook up the power line from the utility pole (marked with a circle on the diagram) to the utility box (marked with a star). The pole is 300 ft from one edge of the building, which is 1500 ft long. The connector is located at the other edge. The utility company must use a higher grade of cable to connect to the house, and can run a lower grade of line along the house. If the higher grade of cable costs $.10 per foot and the cheaper grade costs $.07 per foot, where should the cable connect to the house to minimize the cost of the cable?

I am just unable to even understand where to begin. I know for optimization problems, the first step is to get an equation, then take its derivative, and then set that equal to 0. However, I can't even formulate an equation.

It seems to me that it's cheapest to just directly connect the cable from the connector straight to the house so the house (300ft), then run it along the house (1500ft) and the total cost is $135.00

 

[berni2723]

well, for starters, your assumption that that would be the cheapest way is a little off. if you increase the angle, the amount of overall cable will be less (as the hypotenuse is shorter than 2 sides) thereby spending more on the connecting cable, yet if optimized, saving even more on the running cable.

to solve this here is what you should look for:

1) an equation of cost

2)an equation of distance (note: you have a right angle with sides of 300ft and 1500ft)

then, as you correctly said, solve one equation for 1 variable, substitute, then find the derivative and set it =0 .

good luck (test tomorrow?)

 

[berni2723]

and just one more tip,

NOTE: i solved this, and calling the area from where the cable hits the house to the converter "y" and the rest of the house (where there is no cable) "1500-y" was a good idea

 

[SittsK]

Thank you so much for responding at such a late time! So I figured out the equation of cost to be .1x+.07(1500-y) and by solving for x I got x=.7y-1050. Am I on the right track? I'm still a little confused on the distance equation. So using the Pythagorean Theorem (300^2+1500^2=c^2) I got c=300*sqrt(26). Now i'm stumped.

 

[berni2723]

youre on the right track, but based on the fact that you are calling the wire on the house 1500-y, the P.T. you should use is 300²-y²=x², then solve for x in this one......because you always need to set the derivative of the cost equation to 0, because its the one you want to optimize

 

[SittsK]

Got it!!!! Thank you ^_^