Fred is making cupcakes for a charity sale. The ingredients cost 15p per cake. Suppose that each cake is sold for Β£πΆ and that Fred sells π cakes.
a) Write down a formula for the total profit Β£π Fred makes, in terms of πΆ and π.
$P = (C - 0.15)n$
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 π = β20πΆ + 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 $π = β20πΆ^2 + 78πΆ β 11.25$
$P = (C - 0.15)n\\ => P = (C - 0.15)(-20C + 75)\\ => π = β20πΆ^2 + 75πΆ +3C β 11.25\\ => π = β20πΆ^2 + 78πΆ β 11.25 $
Seemed a bit simple, have I missed a more elegant solution?
d) By expanding $(πΆ β 1.95)^2$, find a number π such that $π = β20(πΆ β 1.95)^2 + k$
$(πΆ β 1.95)^2 = C^2 - 3.9C + 3.8025\\ => π = β20(C^2 - 3.9C + 3.8025) + k\\ => π = β20C^2 + 78C - 76.05 + k$
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 $π = β20πΆ^2 + 78πΆ β 11.25$
Using the maximum value of a quadratic equation max = c - (b2 / 4a):
$=> max = -11.25 - (78^2 / 4(-20))\\ => max = 64.80$
$=> β20πΆ^2 + 78πΆ β 11.25 = 64.80\\ => β20πΆ^2 + 78πΆ β 76.05 = 0 $
Using Quadratic Formula: C = 1.95
So Fred should charge Β£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)?