# Determine the total profit made on the sale of cupcakes, as a formula based on sale price and number sold.

Fred is making cupcakes for a charity sale. The ingredients cost 15p per cake. Suppose that each cake is sold for £𝐶 and that Fred sells 𝑛 cakes.

a) Write down a formula for the total profit £𝑃 Fred makes, in terms of 𝐶 and 𝑛.

$$P = (C - 0.15)n$$

90 people will attend the charity sale. If Fred charges nothing at all for his cakes, 75 people will take a cake (and Fred will make no money). For each pound above zero Fred charges per cake, 20 fewer people will buy a cake.

b) Explain why 𝑛 = −20𝐶 + 75

If C=0, n = 75
If C=1, n = 75 - 20
=> n = 75 - 20C, i.e. n = -20C + 75

Is that enough of an explanation, doesn't seem adequate?

c) Show that $$𝑃 = −20𝐶^2 + 78𝐶 − 11.25$$

$$P = (C - 0.15)n\\ => P = (C - 0.15)(-20C + 75)\\ => 𝑃 = −20𝐶^2 + 75𝐶 +3C − 11.25\\ => 𝑃 = −20𝐶^2 + 78𝐶 − 11.25$$

Seemed a bit simple, have I missed a more elegant solution?

d) By expanding $$(𝐶 − 1.95)^2$$, find a number 𝑘 such that $$𝑃 = −20(𝐶 − 1.95)^2 + k$$

$$(𝐶 − 1.95)^2 = C^2 - 3.9C + 3.8025\\ => 𝑃 = −20(C^2 - 3.9C + 3.8025) + k\\ => 𝑃 = −20C^2 + 78C - 76.05 + k$$

From part (d)
$$-11.25 = -76.05 + k\\ => k = 64.8$$

e) What price should Fred charge for each cupcake if he wishes to raise the most money possible for charity? Explain your answer.

This should be the maximum value of $$𝑃 = −20𝐶^2 + 78𝐶 − 11.25$$

Using the maximum value of a quadratic equation max = c - (b2 / 4a):

$$=> max = -11.25 - (78^2 / 4(-20))\\ => max = 64.80$$

$$=> −20𝐶^2 + 78𝐶 − 11.25 = 64.80\\ => −20𝐶^2 + 78𝐶 − 76.05 = 0$$

Using Quadratic Formula: C = 1.95

So Fred should charge £1.95 to raise the most money for charity.

Is there a quicker, more elegant, method to come to this conclusion (if it's correct ofc)?

• Is it OK to use calculus? If so there's a faster way. Dec 29, 2021 at 10:50
• @coffeemath Precalculus at the moment :) Dec 29, 2021 at 11:10

Part e follows immediately from part d since you have gotten the quadratic into vertex form with vertex at $$C=1.95$$, which must correspond with the max profit.