Question

I need this question solved with derivatives please:

A rectangular piece of cardboard, 100 cm by 40 cm, is going to be used to
make a rectangular box with an open top by cutting congruent squares from
the corners. Calculate the dimensions (to one decimal place) for a box with
the largest volume.

Answer 1

16,242. 7 cm^3.

Explanation:

We need to cut off a square piece at the 4 corners of the cardboard.

Let the length of their edges be x cm.

The volume of the box will be:

V = height * width * length

V = x(100-2x)(40-2x)

V = x(4000 - 200x - 80x + 4x^2)

V = x(4x^2 - 280x + 4000)

V = 4x^3 + - 280x^2 + 4000x

Finding the derivative:

dV / dx = 12x^2 - 560x + 4000 = 0 ( when V is a maxm or minm.)

4(3x^2 - 140x + 1000) = 0

x = 37.86, 8.80.

Looks like x = 8.80 is the right value but we can check this out be looking at the sign of the second derivative:

V" = 24x - 560, when x = 8.8 V" is negative so this is a Maximum for V.

So the maximum volume of the box is when x = 8.8 so we have

V = 8.8(100-2(8.8)(40 - 2(8.8)

= 16,242. 7 cm^3.