SOLUTION: Zach is building a new shed shaped like a square prism. He wants the height of the shed to be 2 feet less than the length and width. If he needs the volume to be as close as poss

Question 1065297: Zach is building a new shed shaped like a square prism. He wants the
height of the shed to be 2 feet less than the length and width. If he needs the volume to
be as close as possible to 3174 ft^3, what should the length be? Round to the nearest foot.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
square prism means the base is a square.

the volume of a square prism is equal to s^2 * h.

s is the length of a side of the base.
h is the height.

he wants the height to be 2 feet less than the length and the width.

if the side length is equal to s, then the height must be equal to s-2

the volume is therefore equal to s^2 * (s-2).

set that equal to 3174 and your equation is 3174 = s^2 * (s-2)

simplify to get 3174 = s^3 - 2s^2

subtract 3174 from both sides of the equation and you get s^3 - 2s^2 - 3174 = 0.

i don't actually know how to solve that without graphing it, so that's what i did.

my equation to graph is y = x^3 - 2x^2 - 3174.

i looked for the point where that graph crosses y = 0.

that point is (15.394,0)

that's rounded to 3 decimal places, but it's close.

the length of the sides is 15.394 and the height of the sides is 13.394.

15.394^2 * 13.394 = 3174.046311 which is pretty close.

here's my graph:

$$$

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