A square sheet of metal is 18 cm by 18 cm. A square is cut from each… Step by Step

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The question

Question: A square sheet of metal is 18 cm by 18 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 18 − 2x on each side.

x = the side of each cut square

Each side loses x at both ends: 18 minus x minus x.

18 − x − x = 18 − 2x

2 Write the volume

Volume is length times width times height.

V = length × width × height

The base is 18 − 2x by 18 − 2x, and the height is the fold, x.

V = (18 − 2x)(18 − 2x) × x

= x(18 − 2x)²

Multiply out the square first: 324, minus 72x, plus 4x².

(18 − 2x)² = 324 − 72x + 4x²

Then times x.

Now we have V = 4x³ − 72x² + 324x.

= 4x³ − 72x² + 324x

3 Which x make sense

x must be more than 0.

And 18 − 2x must be more than 0, so x is less than 9.

0 < x < 9 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

V′(x) = 4 × 3x^(3 − 1) − 72 × 2x^(2 − 1) + 324 × 1

Now we have V′(x) = 12x² − 144x + 324.

= 12x² − 144x + 324

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 − 144x + 324 = 0

Take out the 12.

12(x² − 12x + 27) = 0

We need two numbers that multiply to 27 and add to −12: −3 and −9.

12(x − 3)(x − 9) = 0

−3 × (−9) = 27, −3 + (−9) = −12

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

a × b = 0 → a = 0 or b = 0

So x = 3 or x = 9.

x = 3 or x = 9

x = 9 is not allowed: the base would be 18 − 18 = 0.

No box at all.

x = 9: 18 − 2 × 9 = 0 ✗

Careful!

Check every x against the allowed ones.

9 cuts the whole sheet away.

Careful: not x = 9 ✗

So x = 3.

x = 3 ✓

5 Make sure it is the biggest

Before 3, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

V′(2) = 84 > 0 ↗, V′(4) = −60 < 0 ↘

6 Answer the question

The base is 18 − 6 = 12.

So the largest volume is 432 cm³.

V(3) = 3 × 12 × 12 = 432 cm³

7 Check

Let's check with x a bit smaller: 2 gives 392.

x = 2: V = 392

And x = 4 gives 400.

Both are smaller than 432.

Right!

x = 4: V = 400

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