How about this hard question?

Question

It is a storage bin. Find the dimension of the largest open-top storage bin with a square base and vertical sides that can be made from 108 ft^2 of sheet steel. It may neglect the thickness of the steel and assume that there is no waste.
This is a calculus problem.

Answer 1

The no waste assumption requires cutting and welding. The length of the sheet must be
L = s + 4h
W = s
A = 108 = s^2 + 4sh
V = hs^2
h = V/s^2
s^2 + V/s = 108
s^3 + V = 108s
V = 108s - s^3
dV/ds = 108 - 3s^2 = 0 for min/max
s^2 = 36
s = 6 ft
36 + 24h = 108
24h = 72
h = 3 ft
V = 108 ft^3

Answer 2

Hi,

If the square base has a side length of x feet, then its area is x² square feet. Since you can use 108 ft² for the base and the 4 sides, each side can have an area of (108 - x²)/4 ft². Each side's base is x feet long, so its height equals its area, (108 - x²)/4, divided by its base, x. That makes the height of each side (108 - x²)/(4x).

So the box has a length, a width, and a height of x, x, and (108 - x²)/(4x) respectively. Its volume is V = x*x*(108 - x²)/(4x). This simplifies to V = (108x - x³)/(4) or 27x - ¼x³.

If I enter Y1 = 27x - ¼x³ into your calculator you can look for the maximum volume that is formed and find its x value for the side length.

If you look at the graph, its local maximum occurs when x = 6 and the maximum volume is 108 ft³.

I hope that helps!! :-)

Answer 3

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Answer 4

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