Question

A box is to be constructed from a sheet of cardboard that is 10 cm by 50 cm by cutting out squares of length x by x from each corner and bending up the sides.
What is the maximum volume this box could have? (Round your answer to two decimal places.)

Answer 1

i got 2.36 is that right?

Answer 2

yes

Answer 3

So the box will have height x, length 10 - 2x, and width 50 - 2x, so we have:
v = x(10 - 2x)(50 - 2x)
v = x(500 - 20x - 100x + 4x^2)
v = 4x^3 - 120x^2 + 500x
dv/dx = 12x^2 - 240x + 500
Set that to zero:
12x^2 - 240x + 500 = 0
x = (-(-240) +/- sqrt((-240)^2 - 4*12*500)) / (2*12)
x = (240 +/- sqrt(57600 - 24000)) / 24
x = (240 +/- sqrt(33600)) / 24
x = 10 +/- sqrt(175/3)
But if x is greater than 5, the length of the box is negative, which doesn't make sense, so x can't be 10 + sqrt(175/3), which is approximately 17.6, so therefore x = 10 - sqrt(175/3) =~ 2.36