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Vladimir [108]
3 years ago
11

QUESTION 3 [10 MARKS] A bakery finds that the price they can sell cakes is given by the function p = 580 − 10x where x is the nu

mber 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)
Mathematics
1 answer:
Harman [31]3 years ago
6 0

Answer:

(a)Revenue function, R(x)=580x-x^2

Marginal Revenue function, R'(x)=580-2x

(b)Fixed cost =900 .

Marginal Cost Function=300+50x

(c)Profit,P(x)=-35x^2+280x-900

(d)x=4

Step-by-step explanation:

<u>Part A </u>

Price Function= 580 - 10x

The revenue function

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

The marginal revenue function

\dfrac{dR}{dx}= \dfrac{d}{dx}(R(x))=\dfrac{d}{dx}(580x-x^2)=580-2x\\R'(x)=580-2x

<u>Part B </u>

<u>(Fixed Cost)</u>

The total cost function of the company is given by c=(30+5x)^2

We expand the expression

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

Therefore, the fixed cost is 900 .

<u> Marginal Cost Function</u>

If  c=900+300x+25x^2

Marginal Cost Function, \frac{dc}{dx}= (900+300x+25x^2)'=300+50x

<u>Part C </u>

<u>Profit Function </u>

Profit=Revenue -Total cost

580x-10x^2-(900+300x+25x^2)\\580x-10x^2-900-300x-25x^2\\$Profit,P(x)=-35x^2+280x-900

<u> Part D </u>

To maximize profit, we find the derivative of the profit function, equate it to zero and solve for x.

P(x)=-35x^2+280x-900\\P'(x)=-70x+280\\-70x+280=0\\-70x=-280\\$Divide both sides by -70\\x=4

The number of cakes that maximizes profit is 4.

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