A square sheet of card is 6 cm by 6 cm. A square is cut from each… Step by Step

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The question

Question: A square sheet of card is 6 cm by 6 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 6 − 2x on each side.

x = the side of each cut square

Each side loses x at both ends: 6 minus x minus x.

6 − x − x = 6 − 2x

2 Write the volume

Volume is length times width times height.

V = length × width × height

The base is 6 − 2x by 6 − 2x, and the height is the fold, x.

V = (6 − 2x)(6 − 2x) × x

= x(6 − 2x)²

Multiply out the square first: 36, minus 24x, plus 4x².

(6 − 2x)² = 36 − 24x + 4x²

Then times x.

Now we have V = 4x³ − 24x² + 36x.

= 4x³ − 24x² + 36x

3 Which x make sense

x must be more than 0.

And 6 − 2x must be more than 0, so x is less than 3.

0 < x < 3 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

V′(x) = 4 × 3x^(3 − 1) − 24 × 2x^(2 − 1) + 36 × 1

Now we have V′(x) = 12x² − 48x + 36.

= 12x² − 48x + 36

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 − 48x + 36 = 0

Take out the 12.

12(x² − 4x + 3) = 0

We need two numbers that multiply to 3 and add to −4: −1 and −3.

12(x − 1)(x − 3) = 0

−1 × (−3) = 3, −1 + (−3) = −4

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

a × b = 0 → a = 0 or b = 0

So x = 1 or x = 3.

x = 1 or x = 3

x = 3 is not allowed: the base would be 6 − 6 = 0.

No box at all.

x = 3: 6 − 2 × 3 = 0 ✗

Careful!

Check every x against the allowed ones.

3 cuts the whole sheet away.

Careful: not x = 3 ✗

So x = 1.

x = 1 ✓

5 Make sure it is the biggest

Before 1, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

V′(0.5) = 15 > 0 ↗, V′(2) = −12 < 0 ↘

6 Answer the question

The base is 6 − 2 = 4.

So the largest volume is 16 cm³.

V(1) = 1 × 4 × 4 = 16 cm³

7 Check

Let's check with x a bit smaller: 0.5 gives 12.5.

x = 0.5: V = 12.5

And x = 2 gives 8.

Both are smaller than 16.

Right!

x = 2: V = 8

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