c, f\"(x) Briefly explain your answers! 2. An airline notes that if the price of
ID: 2864470 • Letter: C
Question
c, f"(x) Briefly explain your answers! 2. An airline notes that if the price of an 8 am round trip ticket from Boston to New York is set at $250, they can sell 1000 tickets. For every $50 increase in price, they can sell 100 fewer tickets. For every $50 decrease in price, they can sell 100 more, up to their capacity of 2000 seats. a. Find the demand function. b. If they charge $250, what is the total consumer surplus? c. Find the price and quantity for which revenue is maximized. 3. The function f(x y) x y 10x-8y +1 has one critical point. Find it, and determine if it is a local minimum, local maximum, or saddle point. 4. Find fa, fo and fx or f(r,y) y ser. Also do the following problems from the textbook Chapter 2 #70, Chapter 3 #48, #98, #116, Chapter 4 #44, 58Explanation / Answer
Let us say the price is increased by 50$, x number of times, then the new price becomes (250+50x) and the new demand becomes (1000-100x). Here, positive x will represent increase in price and negative x will represent decrease in price.
a) Therefore, demand function is D(x) = (1000-100x)
b) If they charge $250, the consumer surplus will be equal to:
(Price the customer is willing to pay) ~ (Price the customer is actually paying)
=(250+50x)-(250) = 50x
Therefore, consumer surplus is 50x.
c) Revenue = Cost*Demand = (250+50x)(1000-100x)
R(x) = 250000-25000x+50000x-5000x^2
R(x) = -5000x^2+25000x+250000
In order to find maximized revenue, we can use derivaties. Let us first find the critical points of this function. For finding critical points, we set R'(x) = 0 and solve for x.
R'(x) = -10000x+25000 = 0
10000x = 25000
x = 2.5
Price = (250+50x) = 250+50*2.5 = 250+125 = $375
Quantity = (1000-100x) = 1000 - 100*2.5 = 1000 - 250 = 750
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