A shop sells 300 cakes a day at $5 each. For every $1 it adds to the… Step by Step

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The question

Question: A shop sells 300 cakes a day at $5 each.

For every $1 it adds to the price, it sells 20 fewer cakes.

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 $5 + x.

price = 5 + x

And each rise loses 20 cakes: they sell 300 − 20x.

cakes = 300 − 20x

2 Write the money

The money is the price times the number sold.

R = price × number sold

R = (5 + x)(300 − 20x)

Multiply out the brackets.

Now we have R = 1500 + 200x − 20x².

= 1500 + 200x − 20x²

3 Which x make sense

x is 0 or more, and they must still sell some cakes: x is less than 15.

0 ≤ x < 15 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

R′(x) = 0 + 200 × 1 − 20 × 2x^(2 − 1)

Now we have R′(x) = 200 − 40x.

= 200 − 40x

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

So we solve R′(x) = 0.

R′(x) = 0

200 − 40x = 0

Take out the −40.

−40(−5 + x) = 0

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.

R′(4) = 40 > 0 ↗, R′(6) = −40 < 0 ↘

6 Answer the question

The best price is $5 + 5: $10.

price = 5 + 5 = $10

Careful!

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

The price is 5 + 5.

Careful: not price = $5 ✗

Then they sell 200 cakes, and take in $2000.

R(5) = 10 × 200 = $2000

7 Check

Let's check with x a bit smaller: 4 gives 1980.

x = 4: R = 1980

And x = 6 gives 1980.

Both are smaller than 2000.

Right!

x = 6: R = 1980

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