Volume question

Question

An open box is to be made out of a 6-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimensions of the bottom of the box:

Height of the box: 

Answer 1

Let x = measure of the side of the square (0<x<3), if x = 3, no box)

The width of the box is then 6 - 2x ( and x cut off each corner)

The length of the box is 18 - 2x

Height is x

The volume of the resulting open box depending on x:

V(x) = (18 - 2x)(6 - 2x) x

= 4(x³ - 12x² + 27x)

V'(x) = 4(3x² - 24x + 27)

The 1st derivative equals zero when:

x² - 8x + 9 = 0

x² - 8x = -9

x² - 8x + 16 = -9 + 16

(x - 4)² = 7

x = 4 ± √7 (reject the plus instance because outside the domain of V)

4 - √7 ≈ 1.35

V'(1) > 0

V'(2) < 0

So V goes from increasing to decreasing at 4 - √7, meaning max.

Height: 4 - √7

Width: 6 - 2(4 - √7) = -2 + 2√7

Length: 18 - 2(4 - √7) = 10 + 2√7

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