Question

You are holding a movie party and want to make your own boxes for popcorn out of some old movie posters, which are 8.5 x 11 inches in size. You will construct the open-top boxes by cutting a square out of each corner of the poster and folding up the sides. Being the generous host that you are, you want to make boxes that hold maximum amount of popcorn.

Answer 1

x = 3,17 in

Explanation:

Movie poster dimensions:

8,5 and 11 inches

Let´s call "x" the side of the square, to cut in each corner, then the volume of the open box will be:

V(ob) = ( 8,5 - x ) * ( 11 - x )* x

Then volume as a function of x

V(x) = ( 8,5 - x ) * ( 11 - x )* x or V(x) = ( 93,5 - 8,5*x - 11*x + x² )*x

V(x) = 93,5*x -19,5*x² + x³

Taking derivatives on both sides of the equation:

V´(x) = 93,5 - 39*x + 3*x²

V´(x) = 0 ⇒ 93,5 - 39*x + 3*x² = 0

3*x² - 39*x + 93,5 = 0

Solving for x

x₁,₂ = ( 39 ± √1521 - 1122 )/6

x₁,₂ = 39 ± 19,97 / 6

x₁ = 19,03/6 x₁ = 3,17 in

x₂ = 58,97 / 6 x₂ = 9,83 dismiss this root (we can´t cut 2 pieces of 9,83 from 8,5 inches

x = 3,17 in