A square sheet of cardboard is 24 in by 24 in. A square is cut from… Step by Step

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The question

Question: A square sheet of cardboard is 24 in by 24 in.

A square is cut from each corner and the sides are folded up to make an open box.

Find the largest volume.

1 Draw and name

Cut an x by x square from each corner.

The base that is left is 24 − 2x on each side.

x = the side of each cut square

Each side loses x at both ends: 24 minus x minus x.

24 − x − x = 24 − 2x

2 Write the volume

Volume is length times width times height.

V = length × width × height

The base is 24 − 2x by 24 − 2x, and the height is the fold, x.

V = (24 − 2x)(24 − 2x) × x

= x(24 − 2x)²

Multiply out the square first: 576, minus 96x, plus 4x².

(24 − 2x)² = 576 − 96x + 4x²

Then times x.

Now we have V = 4x³ − 96x² + 576x.

= 4x³ − 96x² + 576x

3 Which x make sense

x must be more than 0.

And 24 − 2x must be more than 0, so x is less than 12.

0 < x < 12 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

V′(x) = 4 × 3x^(3 − 1) − 96 × 2x^(2 − 1) + 576 × 1

Now we have V′(x) = 12x² − 192x + 576.

= 12x² − 192x + 576

At a turning point the graph is flat for a moment: the slope is 0.

So we solve V′(x) = 0.

V′(x) = 0

12x² − 192x + 576 = 0

Take out the 12.

12(x² − 16x + 48) = 0

We need two numbers that multiply to 48 and add to −16: −4 and −12.

12(x − 4)(x − 12) = 0

−4 × (−12) = 48, −4 + (−12) = −16

A product is 0 only when one of its parts is 0.

a × b = 0 → a = 0 or b = 0

So x = 4 or x = 12.

x = 4 or x = 12

x = 12 is not allowed: the base would be 24 − 24 = 0.

No box at all.

x = 12: 24 − 2 × 12 = 0 ✗

Careful!

Check every x against the allowed ones.

12 cuts the whole sheet away.

Careful: not x = 12 ✗

So x = 4.

x = 4 ✓

5 Make sure it is the biggest

Before 4, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

V′(3) = 108 > 0 ↗, V′(5) = −84 < 0 ↘

6 Answer the question

The base is 24 − 8 = 16.

So the largest volume is 1024 in³.

V(4) = 4 × 16 × 16 = 1024 in³

7 Check

Let's check with x a bit smaller: 3 gives 972.

x = 3: V = 972

And x = 5 gives 980.

Both are smaller than 1024.

Right!

x = 5: V = 980

Let's look back.

First: draw and name.

Then: write the volume.

Then: which x make sense.

Then: slope = 0.

Then: make sure it is the biggest.

Then: answer the question.

Last: check.

Answer