ECN 230 Problem Set 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Problem Set 1 (40 points)
ECN 230
Winter 2023
I set take-home open-book assignments rather than midterm exams because I think it can be a better chance for students to demonstrate their problem-solving skills — so I invest a lot of trust in your academic integrity. In return, I ask you to have enough trust in your own abilities to not work with anyone else or consult the internet. For this reason: please write the honour code on your solutions.
I pledge that I will neither give nor receive any aid from any other person during this assignment. The work presented here is entirely my own.
1. (25 points) You are in charge of selling tickets to a concert. If you post f
flyers and tell n of your friends about the concert, you will sell t(f , n)
total tickets.
a) (2 points) How many first order partial derivatives does the function t have? What are they?
If you post 100 flyers and tell 15 friends, you will sell 200 tickets and
t
(100, 15) = 1 t
(100, 15) = 2
b) (2 points) Choose one of these and explain what it describes, in the context of tickets, flyers, and friends.
The concert hall has 200 seats, so you aim to sell exactly 200 tickets.
c) (1 points) Relate this goal to the concept of a level curve.
d) (3 points) Let n = ℓ(f) be this level curve. Describe what ℓ\ (f) means, in the context of tickets, flyers, and friends.
e) (4 points) Compute ℓ\ (100). Justify your answer mathematically. It takes 15 minutes to post each flyer and 30 minutes to tell each friend about the concert.
f) (1 point) Write the expression representing the total time in minutes it will take you to post f flyers and tell n friends.
g) (3 points) What is the minimum amount of time needed to do the promotion to sell 200 tickets? Justify your answer mathematically, using only methods or concepts you have already seen.
Now suppose you no longer have time to do promotion, so you hire some assistants to put up flyers and email your friends. If you hire a assistants, for h hours each, they will put up f (a, h) flyers and email n(a, h) friends. (The assistants do not necessarily post flyers or email friends at the same speed as you do!)
h) (4 points) Describe what 
represents, in the context of tickets, fly- ers, friends, assistants, and hours.
i) (2 points) Apply the chain rule to find 
.
j) (3 points) What do you expect the sign of 
to be? Explain why, discussing the contribution of each derivative or partial derivative.
2. (15 points) A co-op cooks meals and throws parties. To cook ten meals, the co-op members need to eat two meals themselves and have one members- only cooking party. To throw two parties, the co-op members need to eat six meals while preparing and have one members-only clean-up party. The co-op aims to serve the public dm meals and have dp public parties.
a) (6 points) Represent this economy as a system of equations.
b) (5 points) How many total meals are cooked by co-op members? How many total parties are thrown by the co-op? Justify your answer mathematically.
c) (2 points) How many clean-up parties are thrown? Justify your an- swer mathematically.
d) (2 points) If the co-op increases the number of parties they have for the public, what is the impact on the number of meals cooked? Justify your answer mathematically. Intuitively, what explains this impact?
2023-02-27