A square sheet of tin is 30 cm by 30 cm. A square is cut from each… Step by Step

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The question

Question: A square sheet of tin is 30 cm by 30 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 30 − 2x on each side.

x = the side of each cut square

Each side loses x at both ends: 30 minus x minus x.

30 − x − x = 30 − 2x

2 Write the volume

Volume is length times width times height.

V = length × width × height

The base is 30 − 2x by 30 − 2x, and the height is the fold, x.

V = (30 − 2x)(30 − 2x) × x

= x(30 − 2x)²

Multiply out the square first: 900, minus 120x, plus 4x².

(30 − 2x)² = 900 − 120x + 4x²

Then times x.

Now we have V = 4x³ − 120x² + 900x.

= 4x³ − 120x² + 900x

3 Which x make sense

x must be more than 0.

And 30 − 2x must be more than 0, so x is less than 15.

0 < x < 15 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

V′(x) = 4 × 3x^(3 − 1) − 120 × 2x^(2 − 1) + 900 × 1

Now we have V′(x) = 12x² − 240x + 900.

= 12x² − 240x + 900

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 − 240x + 900 = 0

Take out the 12.

12(x² − 20x + 75) = 0

We need two numbers that multiply to 75 and add to −20: −5 and −15.

12(x − 5)(x − 15) = 0

−5 × (−15) = 75, −5 + (−15) = −20

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

a × b = 0 → a = 0 or b = 0

So x = 5 or x = 15.

x = 5 or x = 15

x = 15 is not allowed: the base would be 30 − 30 = 0.

No box at all.

x = 15: 30 − 2 × 15 = 0 ✗

Careful!

Check every x against the allowed ones.

15 cuts the whole sheet away.

Careful: not x = 15 ✗

So x = 5.

x = 5 ✓

5 Make sure it is the biggest

Before 5, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

V′(4) = 132 > 0 ↗, V′(6) = −108 < 0 ↘

6 Answer the question

The base is 30 − 10 = 20.

So the largest volume is 2000 cm³.

V(5) = 5 × 20 × 20 = 2000 cm³

7 Check

Let's check with x a bit smaller: 4 gives 1936.

x = 4: V = 1936

And x = 6 gives 1944.

Both are smaller than 2000.

Right!

x = 6: V = 1944

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