FALL 2015
MATH 141 - ASSIGNMENT (1.3, 1.4, 1.Q)
Name
1. A fast food outlet has recently added a new burrito to their menu. Based on market research, they
found that the monthly demand for this kind of burrito is 600 when the unit price is $4, and the
monthly demand is 300 when the unit price is $6. The cost of making each burrito is $1. The
monthly fixed cost is $1500.
(i) (5pts) Find the demand equation (assuming it is linear).
(ii) (2pts) Find the revenue function.
(iii) (3pts) Find the profit function.
(iv) (5pts) Determine the break-even point. What is the maximum number of burritos they can
make and sell in a month without making a loss?
(v) (5pts) How many burritos must be sold each month to maximize revenue? What is the
maximum revenue?
(vi) (5pts) How many burritos must be sold each month to maximize profit? What is the
maximum profit?
Solution:
(i) Let p be the unit price in $. Let x be the demand.
∆p
6−4
1
= 300−600
= − 150
.
m = ∆x
1
1
p = mx + b = − 150 x + b. Plugging in x = 300 and p = 6 gives 6 = − 150
(300) + b so b = 8.
1
The demand equation is p = − 150 x + 8.
1
1 2
(ii) R(x) = x · p(x) = x(− 150
x + 8) = − 150
x + 8x.
(iii) The cost function is C(x) = x + 1500.
1 2
1 2
The profit function is P (x) = R(x) − C(x) = (− 150
x + 8x) − (x + 1500) = − 150
x + 7x − 1500.
(iv) R(x) = C(x) (or equivalently, P (x) = 0). Solving for x gives x = 300 or x = 750. Plugging
in x = 300 into the cost function gives C(300) = 300 + 1500 = 1800. The break-even point is
(300 burritos, $1800).
They can make and sell at most 750 burritos in a month without making a loss.
b
8
= 600. Plugging this into the
(v) We want the vertex of the revenue function. x = − 2a
= − −2/150
1
2
revenue function gives R(600) = − 150 (600 ) + 8(600) = 2400.
600 burritos must be sold to get a maximum revenue of $2400.
b
7
(vi) We want the vertex of the profit function. x = − 2a
= − −2/150
= 525. Plugging this into the
1
2
profit function gives P (525) = − 150 (525 ) + 7(525) − 1500 = 337.5.
525 burritos must be sold to get a maximum profit of $337.50.
1