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.
Find the revenue and marginal revenue functions [Hint: revenue is price multiplied by quantity i.e. revenue = price × quantity] (3 marks)

Working and Answer Space for Part A

Find the fixed cost and marginal cost function [Hint: fixed cost does not change with quantity produced] (3 marks)

Working and Answer Space for Part

From the information given we know that the price they can sell cakes is given by the function [tex]p=580-10x[/tex], where [tex]x[/tex] is the number of cakes sold per day.

So, the total revenue is

[tex]TR=(580-10x)\cdot x\\TR=580x-10x^2[/tex]

And the marginal revenue is

[tex]MR=\frac{d}{dx}(580x-10x^2) \\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\MR=\frac{d}{dx}\left(580x\right)-\frac{d}{dx}\left(10x^2\right)\\\\MR=580-20x[/tex]

The Fixed Cost ([tex]FC[/tex]) is the amount of money you have to spend regardless of how many items you produce.

The Marginal Cost ([tex]MC[/tex]) function is the derivative of the cost function and is given by

[tex]MC=\frac{d(TC)}{dx}[/tex]

We know that the total cost function of the company is given by [tex]C=(30+5x)^2[/tex], which it is equal to

[tex]\mathrm{Apply\:Perfect\:Square\:Formula}:\quad \left(a+b\right)^2=a^2+2ab+b^2\\a=30,\:\:b=5x\\\\\left(30+5x\right)^2=30^2+2\cdot \:30\cdot \:5x+\left(5x\right)^2=25x^2+300x+900\\\\C=25x^2+300x+900[/tex]

From the total cost function and applying the definition of fixed cost, the fixed cost is $900.

And the marginal cost function is

[tex]MC=\frac{d}{\:dx}\left(25x^2+300x+900\right)\\\\MC=\frac{d}{dx}\left(25x^2\right)+\frac{d}{dx}\left(300x\right)+\frac{d}{dx}\left(900\right)\\\\MC=50x+300+0=50x+300[/tex]