Two buildings at opposite corners of a parking lot need to be connected by cable that will be buried under ground. It costs $11/ft to lay accross parking lot and $8/ft to lay along the sides. The lot is 300ft by 360 ft rect from pt A to the opposite corner with indents of 60ft off each side. What is cheapest path?

Question

Two buildings at opposite corners of a parking lot need to be connected by cable that will be buried under ground. It costs $11/ft to lay accross parking lot and $8/ft to lay along the sides. The lot is 300ft by 360 ft rect from pt A to the opposite corner with indents of 60ft off each side. What is cheapest path?

Answer 1

if the cable goes from A along the 360-ft side for a distance x past the indent at A's corner to y from the indent at B's corner, then the cost will be

8(60+x) + 11√((300-x)^2+(240-y)^2) + 8(60+y)

Now, I don't think you have studied multi-variable derivatives, so there must be some other relationship between x and y.

Answer 2

To find the cheapest path for laying the cable underground, we need to compare the costs of laying the cable across the parking lot with the cost of laying it along the sides.

Let's break down the two options:

1. Laying the cable across the parking lot: This forms a diagonal line connecting the two buildings, resulting in a hypotenuse of a right triangle. The length of the hypotenuse can be found using the Pythagorean theorem: c^2 = a^2 + b^2. Here, a and b represent the lengths of the sides of the rectangle. So, a = 300 ft and b = 360 ft.

Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
c^2 = 300^2 + 360^2
c^2 = 90000 + 129600
c^2 = 219600
c ≈ 468.491 ft

The cost of laying the cable across the parking lot would be $11 per foot, so the total cost would be:
Total Cost = Length of Hypotenuse * Cost per foot
Total Cost = 468.491 ft * $11/ft
Total Cost ≈ $5,153.40

2. Laying the cable along the sides: In this case, we need to consider the length of all four sides of the rectangle. The bottom and top sides both have a length of 300 ft, and the left and right sides both have a length of 360 ft.

The total cost of laying the cable along the sides can be calculated as:
Total Cost = 2 * Length of Bottom/Top Side * Cost per foot + 2 * Length of Left/Right Side * Cost per foot
Total Cost = 2 * 300 ft * $8/ft + 2 * 360 ft * $8/ft
Total Cost = $4,800 + $5,760
Total Cost = $10,560

Therefore, the cheapest path for laying the cable underground is to lay it along the sides of the parking lot, which would cost $10,560.