Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Question

b) Suppose that in part a the original piece of cardboard is a square with side lengths k. Find the volume of the largest box that can be formed in this way.

c)Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length k and L. Holding k fixed, find the size of the corner squares x that maximizes the volume of the box as L→∞.

I solved part a, for x to be approximately .5657 and the volume at approximately 3.03. On part b and c, I am not even sure where to begin.

Answer 1

in inches,
v = (36-2x)(48-2x)x
dv/dx = 12(x^2-28x+144)
dv/dx=0 when x = 6.789" ≈ 0.5657 ft

For (b), just plug in k, and you have
v = x(k-2x)^2
dv/dx = 12x^2 - 8kx + k^2
dv/dx=0 when x = k(2±√3)/6
The max will occur at x = k(2-√3)/6, since (2+√3)/6 > 1/2, and the corners would use up the whole square.

For (c), work the same way, but
v = x(k-2x)(L-2x)

Answer 2

thank you so much steve. great help