Calculus exercise (James Stewart book) : Volume of a box

Question

I'm learning calculus but I'm very stuck in this exercise.

Exercise

I know that the volume is : 2HL + 2HW + LW (only LW because open top).

Thanks to the exercise we know that:

W = 12 - 2x L = 20 - 2x H = x

Consequently I replaced the values:

2x(20 - 2x) + 2x(12 - 2x) + (20 - 2x)*(12 - 2x)

= -4x^2 + 240

My function is f(x) = -4x^2 + 240

However, I don't understand why in the correction it is written 4x^3 - 64x^2 +240x ?

Correction

Answer 1

You haven't calculated the volume of the box, but the surface of the cutout.

Answer 2

As the comment shows, the length of the horizontal box's basis is $\;20-2x\;$ and the vertical one is $\;12-x\;$, so the base's area is $\;(20-2x)(12-2x)=4x^2-64x+240\;$, and since the height is $\;x\;$ your volume function is

$$f(x)=x(4x^2-64x+240)=4x^3-64x^2+240x$$