Question

A bakery finds that the price they can sell cakes is given by the function p = 580 − 10x where x is the number of cakes sold per day, and p is price. The total cost function of the company is given by c = (30+5x) 2 where x is previously defined, and c is total cost.

A. Find the revenue and marginal revenue functions [Hint: revenue is price multiplied by quantity i.e. revenue = price × quantity] (3 marks) B. Find the fixed cost and marginal cost function [Hint: fixed cost does not change with quantity produced] (3 marks) C. Find the profit function [Hint: profit is revenue minus total cost] (2 marks) D. Find the quantity that maximizes profit (2 marks)

Answer 1

A.

The revenue function

$(580-10x)\cdot x=580x-x^2$

The marginal revenue function

$(580x-10x^2)'=580-2x$

B.

The fixed cost

$(30+5x)^2=900+300x+25x^2\\$

The fixed cost 900

The marginal cost function

$(900+300x+25x^2)'=300+50x$

C.

The profit function

$580x-10x^2-(30+5x)^2=-35x^2+280x-900\\$

D.

 The quantity that maximizes profit

$(-35x^2+280x-900)'=-70x+280\\\\-70x+280=0\\ x=4$