A shop sells 120 hats a month at $8 each. For every $1 it adds to… Step by Step

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The question

Question: A shop sells 120 hats a month at $8 each.

For every $1 it adds to the price, it sells 5 fewer hats.

What price gives the most money?

1 Draw and name

Call the number of one-dollar rises x.

x = the number of $1 rises

Then the price is $8 + x.

price = 8 + x

And each rise loses 5 hats: they sell 120 − 5x.

hats = 120 − 5x

2 Write the money

The money is the price times the number sold.

R = price × number sold

R = (8 + x)(120 − 5x)

Multiply out the brackets.

Now we have R = 960 + 80x − 5x².

= 960 + 80x − 5x²

3 Which x make sense

x is 0 or more, and they must still sell some hats: x is less than 24.

0 ≤ x < 24 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

R′(x) = 0 + 80 × 1 − 5 × 2x^(2 − 1)

Now we have R′(x) = 80 − 10x.

= 80 − 10x

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

So we solve R′(x) = 0.

R′(x) = 0

80 − 10x = 0

Take out the −10.

−10(−8 + x) = 0

So x = 8.

x = 8

5 Make sure it is the biggest

Before 8, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

R′(7) = 10 > 0 ↗, R′(9) = −10 < 0 ↘

6 Answer the question

The best price is $8 + 8: $16.

price = 8 + 8 = $16

Careful!

x is the number of one-dollar rises, not the price.

The price is 8 + 8.

Careful: not price = $8 ✗

Then they sell 80 hats, and take in $1280.

R(8) = 16 × 80 = $1280

7 Check

Let's check with x a bit smaller: 7 gives 1275.

x = 7: R = 1275

And x = 9 gives 1275.

Both are smaller than 1280.

Right!

x = 9: R = 1275

Let's look back.

First: draw and name.

Then: write the money.

Then: which x make sense.

Then: slope = 0.

Then: make sure it is the biggest.

Then: answer the question.

Last: check.

Answer