Hello, I am having some trouble finding the functions of this problem in particu

Question

Hello, I am having some trouble finding the functions of this problem in particular. In this situation, what would be the three functions and how do I get them??

Hercules Films is deciding on the price of the video release of its film Bride of the Son of Frankenstein, Marketing estimates that at a price of p dollars, it can sell q = 220 000-10 000p copies, but each copy costs $4 to make (a) Find the cost, revenue and profit functions, C(o)- R(p)- Po)- (b) What price will give the greatest pronit? x dollars Second derivative test Your answer above is a critical point for the profit function. To show it is a maximum, calculate the second derivative of the profit function. P(p)- Evaluate p at your critical point. The result is Select ? X , which means that the profit is Select , X at the critical point, and the critical point is a maximum.

Explanation / Answer

a) We have given q=220,000-10,000p

Cost C(p)=$4*q=4*(220,000-10,000p)=880,000-40,000p

C(p)=880,000-40,000p

Revenue R(p)=p*q

R(p)=p*(220,000-10,000p)

R(p)=220,000p-10,000p^2

Profit function P(p)=R(p)-C(p) =(220,000p-10,000p^2)-(880,000-40,000p)

P(p)=260,000p-10,000p^2-880,000

b) To obtain the maximum of this profit function we take derivative P(p) and set P'(p)=0

P(p)=260,000p-10,000p^2-880,000

P'(p)=260,000-2*10,000p

P'(p)=260,000-20,000p

P'(p)=0

260,000-20,000p=0

-20,000p=-260,000

p=260,000/20,000=13

p=13 dollars

second derivative test:

P''(p)=-20,000

at p=13,P''(13)=-20,000

plug p=13 into profit function P(p)

P(13)=260,000*13-10,000(13)^2-880,000=$810,000

p=$13 gives the greatest profit

at p=13,P''(13)=-20,000,which means that the profit is $810,000 at the critical point, and the critical point is a maximum.

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