Question

If 1,200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

Answer:

4,000 cm³

Step-by-step explanation:

Let x be the length of the sides of the base and h be the height of the box.

The surface area is given by:

$A=x^2+4xh=1,200\\h=\frac{1,200-x^2}{4x}$

The volume if the box is:

$V=hx^2\\V=({\frac{1,200-x^2}{4x}} )*x^2\\V=300x-\frac{x^3}{4}$

The value of x for which the derivate of the function above is zero will produce the largest possible volume:

$V=300x-\frac{x^3}{4} \\V'=300-\frac{3}{4}x^2=0\\ x^2=400\\x=20\ cm$

The height of the box is:

$h=\frac{1,200-20^2}{4*20}\\h=10\ cm$

The largest possible volume is:

$V=10*20^2\\V=4,000\ cm^3$

The largest possible volume of the box is 4,000 cm³

Answer 2

The volume of the open-top box is the amount of space in it.

The largest possible volume of the box is 4000 cubic centimeters

The surface area of an open-top box is:

$\mathbf{A = 2lh + 2wh + lw}$

So, we have:

$\mathbf{2lh + 2wh + lw = 1200}$

The box has a square base.

So, we have:

$\mathbf{l= w }$

The expression becomes

$\mathbf{2lh + 2lh + l^2 = 1200}$

$\mathbf{4lh + l^2 = 1200}$

Make  the subject

$\mathbf{h = \frac{1200 - l^2}{4l}}$

The volume of the box is:

$\mathbf{V = lwh}$

So, we have:

$\mathbf{V = l^2h}$

Substitute $\mathbf{h = \frac{1200 - l^2}{4l}}$

$\mathbf{V = l^2 \times \frac{1200 - l^2}{4l}}$

$\mathbf{V = l \times \frac{1200 - l^2}{4}}$

Expand

$\mathbf{V = \frac{1200l - l^3}{4}}$

Split

$\mathbf{V = 300l - \frac{l^3}{4}}$

Differentiate

$\mathbf{V' = 300 - \frac{3l^2}{4}}$

Set to 0

$\mathbf{ 300 - \frac{3l^2}{4} = 0}$

Rewrite as:

$\mathbf{ \frac{3l^2}{4} = 300}$

Multiply through by 4

$\mathbf{ 3l^2 = 1200}$

Divide both sides by 3

$\mathbf{l^2 = 400}$

Take square roots of both sides

$\mathbf{l = 20}$

Recall that:

$\mathbf{h = \frac{1200 - l^2}{4l}}$

So, we have:

$\mathbf{h = \frac{1200 - 20^2}{4 \times 20}}$

$\mathbf{h = \frac{1200 - 400}{80}}$

$\mathbf{h = \frac{800}{80}}$

$\mathbf{h = 10}$

Recall that:

$\mathbf{V = l^2h}$

So, we have:

$\mathbf{V=20^2 \times 10}$

$\mathbf{V=4000}$

Hence, the largest possible volume of the box is 4000 cubic centimeters

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