A square sheet of card is 12 cm by 12 cm. A square is cut from each… Step by Step

Step by step

The question

Question: A square sheet of card is 12 cm by 12 cm.

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 12 − 2x on each side.

x = the side of each cut square

Each side loses x at both ends: 12 minus x minus x.

12 − x − x = 12 − 2x

2 Write the volume

Volume is length times width times height.

V = length × width × height

The base is 12 − 2x by 12 − 2x, and the height is the fold, x.

V = (12 − 2x)(12 − 2x) × x

= x(12 − 2x)²

Multiply out the square first: 144, minus 48x, plus 4x².

(12 − 2x)² = 144 − 48x + 4x²

Then times x.

Now we have V = 4x³ − 48x² + 144x.

= 4x³ − 48x² + 144x

3 Which x make sense

x must be more than 0.

And 12 − 2x must be more than 0, so x is less than 6.

0 < x < 6 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

V′(x) = 4 × 3x^(3 − 1) − 48 × 2x^(2 − 1) + 144 × 1

Now we have V′(x) = 12x² − 96x + 144.

= 12x² − 96x + 144

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 − 96x + 144 = 0

Take out the 12.

12(x² − 8x + 12) = 0

We need two numbers that multiply to 12 and add to −8: −2 and −6.

12(x − 2)(x − 6) = 0

−2 × (−6) = 12, −2 + (−6) = −8

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

a × b = 0 → a = 0 or b = 0

So x = 2 or x = 6.

x = 2 or x = 6

x = 6 is not allowed: the base would be 12 − 12 = 0.

No box at all.

x = 6: 12 − 2 × 6 = 0 ✗

Careful!

Check every x against the allowed ones.

6 cuts the whole sheet away.

Careful: not x = 6 ✗

So x = 2.

x = 2 ✓

5 Make sure it is the biggest

Before 2, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

V′(1) = 60 > 0 ↗, V′(3) = −36 < 0 ↘

6 Answer the question

The base is 12 − 4 = 8.

So the largest volume is 128 cm³.

V(2) = 2 × 8 × 8 = 128 cm³

7 Check

Let's check with x a bit smaller: 1 gives 100.

x = 1: V = 100

And x = 3 gives 108.

Both are smaller than 128.

Right!

x = 3: V = 108

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