Problem 1. Recall the county of Soggy Bottom from class. It is a square region that runs
80 kilometers East-West and 80 kilometers North-South. The contour diagram below shows
the number of centimeters of rainfall R(x, y) that fell at different points in the county during
the month of April. The numbers x and y indicate the distances (in kilometers) east and
north of the town of Rainbow Reel.
(a) A freeway runs in a straight line between the points (−60, 40) and (0, −20). What
point on the freeway had the most amount of rain and what point had the least
amount of rain? Explain.
(b) A county road runs along the line y = 2x + 60. What point on the county road has
the most amount of rain and what point had the least amount of rain? Explain.
−60 −40 −20 0. 20
−20
0.
20
40
60
Distance East of Rainbow Reel, x (km)
Distance North of Rainbow Reel,
y (km)
Rainbow Reel
5
6
7
8
9
10
6 5
4
7
8
9
Problem 2. A firm manufactures a commodity at two different factories. The total cost
of manufacturing depends on the quantities q1 and q2 manufactured at each factory and is
expressed by the joint cost function
C = 2q
2
1 + q1q2 + q
2
2 + 500.
The company’s objective is to produce 200 units, while minimizing production costs.
(a) How many units should be manufactured at each factory?
(b) Suppose the company decides to produce 210 units instead of 200 units. By about
how much will the production costs increase?
Problem 3. A person has a budget of $120 to spend on videos and cds. Her utility (think
of this as a measure of happiness) from renting x movies and buying y cds is
U(x, y) = x
2/3
y
1/3
It costs $2 to rent a movie and $10 to buy a cd.
(a) How many movies should she rent and how many cds should she buy to maximize
her utility?
(b) Suppose she wins $5 in the lottery. About how much will her utility increase?