A shop sells 100 shirts a week at $20 each. For every $1 it adds to… Step by Step

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The question

Question: A shop sells 100 shirts a week at $20 each.

For every $1 it adds to the price, it sells 2 fewer shirts.

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 $20 + x.

price = 20 + x

And each rise loses 2 shirts: they sell 100 − 2x.

shirts = 100 − 2x

2 Write the money

The money is the price times the number sold.

R = price × number sold

R = (20 + x)(100 − 2x)

Multiply out the brackets.

Now we have R = 2000 + 60x − 2x².

= 2000 + 60x − 2x²

3 Which x make sense

x is 0 or more, and they must still sell some shirts: x is less than 50.

0 ≤ x < 50 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

R′(x) = 0 + 60 × 1 − 2 × 2x^(2 − 1)

Now we have R′(x) = 60 − 4x.

= 60 − 4x

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

So we solve R′(x) = 0.

R′(x) = 0

60 − 4x = 0

Take out the −4.

−4(−15 + x) = 0

So x = 15.

x = 15

5 Make sure it is the biggest

Before 15, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

R′(14) = 4 > 0 ↗, R′(16) = −4 < 0 ↘

6 Answer the question

The best price is $20 + 15: $35.

price = 20 + 15 = $35

Careful!

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

The price is 20 + 15.

Careful: not price = $15 ✗

Then they sell 70 shirts, and take in $2450.

R(15) = 35 × 70 = $2450

7 Check

Let's check with x a bit smaller: 14 gives 2448.

x = 14: R = 2448

And x = 16 gives 2448.

Both are smaller than 2450.

Right!

x = 16: R = 2448

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