Please Help ASAP

Question

Question 1

Given the given cost function

C(x)=8950+720x+0.7x^2 and the demand function p(x)=2160.

Find the production level that will maximaze profit.

Question 2

A baseball team plays in a stadium that holds 52000 spectators. With the ticket price at $9 the average attendence has been 23000. When the price dropped to $7, the average attendence rose to 26000. Assume that attendence is linearly related to ticket price. What ticket price would maximize revenue?

Question 3

A 320-unit hotel is trying to determine their daily rates. From experience, the hotel manager knows that all rooms are typically occupied when the daily room rate is $89. Market research indicates that for every increase of x dollars in the daily room rate, there will be x rooms vacant. Each occupied room costs $22 per day to service and maintain. What should the hotel charge per day in order to maximize profit?

Answer 1

R(x) = 2160x

revenue = price times output.

it's a competitive market

the firm is a price taker

C(x) = 8950+ 720x+ .7x^2

Profit = P(x) = R(x)- C(x)

=-.7x^2+ 1440x -8950

take the derivative and set equal to zero

P'(x) = R'(x) - C'(x) = 0

solve for x = profit maximizing output level

1.4x=1440

x= 14,400/14 = 1,028 4/7

(23,9), (26, 7)= (q,p) where q = quantity measured in thousands of people, p = price in dollars

equation is:

y-9 = (-2/3)(x-23)

y = -2x/3 +46/3+27/3

y= -2x/3 +73/3

or

p=-2q/3 +73/3

revenue = pq = -2q^2/3 +73q/3

R'(q) = -4q/3 +73/3= 0

q = 73/4= 18.25

p = -2(73/4)/3 +73/3

= 73/6=about $12.17

with 18,250 people