EXAM IV
(20 pts) 1. Solve the systems of
equations:
a. 3x + 2y = 10
b. y = x4 - 2x2 + 1
2x + 5y = 3
y = 1 - x2
(10 pts) 2. Maximize the expression z = 3x + 4y subject to the constraints whose set of feasible solutions has the vertices (2,2), (4,0), (7,1), and (5,3).
(15 pts) 3. Ten gallons of a 30% solution is to be formed by mixing a 20% solution with a 50% solution. Find how many gallons of each solution are required.
(10 pts) 4. Find the consumer surplus
for the demand and supply equations
p = 50 - 0.5x
p = 0.125x
(15 pts) 5. Find the equation of a parabola y = ax2 + bx + c which contains the points (1,2), (2,1), and (3,-4).
(15 pts) 6. Sketch the set of feasible
solutions to the set of inequalities, including vertices:
y >= 0
x + y >= 30
x + 2y <= 40
2x + 3y >= 72
(10 pts) 7. Translate this problem into an algebra problem, but DO NOT SOLVE IT:
A car rental agency has just spent $1 million on new cars in three sizes: compact, intermediate, and full-sized. The price of a compact car was $8000; of an intermediate, $12,000; and of a full-sized, $16,000. A total of 100 new cars were bought, and there were twice as many compact cars purchased as intermediate ones. Find how many were purchased of each type.
(10 pts) 8. Translate this problem into an algebra problem, but DO NOT SOLVE IT:
A manufacturer wants to maximize the profit for two products. The first product has a profit of $1.50 for each unit manufactured and sold; the second, $2.00 per unit. The total production level can not exceed 1200 units per month. The demand for product II is less than or equal to half the demand for product I. The production level of product I is less than or equal to 600 units plus three times the production level of product II. Find how many units of each product should be produced in order to maximize the profit.