Answer:
- front/back: 106 2/3 ft
- sides: 135 ft
Step-by-step explanation:
These problems are easily solved if you start with the knowledge that the solution makes the front/back cost equal to the side cost.
Suppose we define the length of the front as x. Then the total cost of the front and back is (2x)(81) = 162x.
If y is the length of the side of the building, then (2y)(64) = 128y is the total cost of the sides of the building. When these costs are equal, we have ...
162x = 128y
y = (162/128)x
The floor area is ...
xy = 14400 = x(162/128)x
x = √(14400·128/162) = √(11377 7/9) = 106 2/3
y = (162/128)x = 135
The front/back of the building measure 106 ft 8 inches; the sides measure 135 feet.
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<em>Solution using derivatives</em>
Using the above variable definitions, we can find the side length as ...
y = 14400/x
so the total cost is then ...
c = 162x + 128(14400/x)
We want the derivative with respect to x to be zero:
dc/dx = 0 = 162 -128·14400/x^2
Solving for x gives ...
x = √(14400·128/162) = 106 2/3 . . . . . compare to the solution above
y = 14400/(106 2/3) = 135
