# data[].attributes.body
In this problem, our goal in this problem was to find the most cost effective way to install a light fixture on a local golf course. We managed to find out that the minimum cost required to install the light is $11562.15

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Total cost= Total cost using trench digging device + Total cost using horizontal boring machine

f(x)= 15(400-x) + 40 √(x^2 + 150^2)

f(x)= 6000 - 15x + 40 √(x^2 +150^2)

f '(x)= 0 - 15 + 40(1/2)(x62 + 150^2)^2 (2x)

f '(x)= (40x)/(√(X^2 + 150^2)) - 15 =0

(15/15) √(x^2 + 150^2) =(40x)/15

(√(150^2 + x^2)^2) = (8/3)x^2

22500 + x^2 = (64/9)x^2

22500 = (55/9)x^2

x = √(3681.818)

x = 60.678

(60.678^2) + (150^2) = c^2

c= $6472.32

400-60.678 = 339.322

339.322 x 15 = $5089.83

$5089.83 + $6472.32 = $11562.15

In order to spend the least amount of money digging a hole through a golf course for a cable to power a light on the other side of the hole without damaging the green, you would need to dig with the trench digging machine ($15 per foot) in a straight line until you are 339.322 feet down the cart path. Then, you would use the horizontal boring machine ($40 per foot) to dig the rest of the way, underneath the green, to the light fixture. If you do that, the total cost to dig would be $11,562.15 which is, in fact, the minimum total cost.

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In this problem, our goal was to find the rate of change of the power factor with the given information. With a couple calculations, we found out the rate of change of the power factor is .012

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A detailed explanation with our calculations and how we found our solution.