Question

Jorge is asked to build a box in the shape of a rectangular prism. The maximum girth of the box is 20 cm. What is the width of the box that would yield the maximum volume? What is the maximum volume given a girth of 20 cm? Use a graphing calculator to determine the width that provides the maximum volume, round to the nearest tenth.

The width of the box is __________cm
The maximum volume is __________cm3

Answer 1

The width of the box is 6.7 cm

The maximum volume is 148.1 cm³

Explanation:

The given parameters of the box Jorge is asked to build are;

The maximum girth of the box = 20 cm

The nature of the sides of the box = 2 square sides and 4 rectangular sides

The side length of square side of the box = w

The length of the rectangular side of the box = l

Therefore, we have;

The girth = 2·w + 2·l = 20 cm

∴ w + l = 20/2 = 10

w + l = 10

l = 10 - w

The volume of the box, V = Area of square side × Length of rectangular side

∴ V = w × w × l = w × w × (10 - w)

V = 10·w² - w³

At the maximum volume, we have;

dV/dw = d(10·w² - w³)/dw = 0

∴ d(10·w² - w³)/dw = 2×10·w - 3·w² = 0

2×10·w - 3·w² = 20·w - 3·w² = 0

20·w - 3·w² = 0 at the maximum volume

w·(20 - 3·w) = 0

∴ w = 0 or w = 20/3 = 6.
`\overline 6`

Given that 6.
`\overline 6` > 0, we have;

At the maximum volume, the width of the block, w = 6.
`\overline 6` cm ≈ 6.7 cm

The maximum volume,
`V_(max)`, is therefore given when w = 6.
`\overline 6` cm = 20/3 cm as follows;

V = 10·w² - w³

`V_(max)` = 10·(20/3)² - (20/3)³ = 4000/27 = 148.
`\overline {148}`

The maximum volume,
`V_(max)` = 148.
`\overline {148}` cm³ ≈ 148.1 cm³

Using a graphing calculator, also, we have by finding the extremum of the function V = 10·w² - w³, the coordinate of the maximum point is (20/3, 4000/27)

The width of the box is;

6.7 cm

The maximum volume is;

148.1 cm³