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MATH 1113
Review for Exam #2 (2.3-3.3)
Solve each system of equations below by either substitution or elimination. State your
solution as an ordered pair.
5 x  y  11
1. 
2 x  3 y  1
 x  6 y  14
2. 
3x  4 y  24
x  3 y  7
3. 
2 x  6 y  11
4. A toy manufacturer has a new product with daily total revenue given by R  35x and
daily total cost given by C  4000  15x . How many units per day must be produced and
sold to give break even for the product?
5. The demand for a certain product is given by 5q  p  450 , and the supply for this product
is given by p  6q  320 , where p is the price in dollars and q is the quantity of items.
Solve the system containing these two equations to find the equilibrium price and the
equilibrium quantity.
6. The supply and demand equations are given for a certain brand of frozen yogurt where q is
the number of half gallons and p is the price per half gallon. The supply equation is
50 p  q  100  0 and the demand equation is 50 p  q  500 . Find the price and quantity that
gives market equilibrium.
7. Joe invested $25,000 in two different mutual funds, one that averages 6% per year and one
that averages 10% per year. At the end of one year, he had earned $2220 in interest. Write a
system of equations and solve it by any method to determine how much he invested in each
fund.
Solve each linear inequality below and state your solution in interval notation.
8. 2(5 x  1)  7 x  8
9. 80 
97  84  71  62  2 x
 90
6
10. A company is producing a new logic board for computers. The annual fixed cost for the
board is $437,500 and the variable cost is $150 per board. If the logic board sells for $500,
write and solve an inequality that gives the number of logic boards that will give a profit for
the product.
11. It is estimated that the annual cost of driving a certain new car is given by the formula
C  0.35m  2200 where m represents the number of miles driven per year and C is the cost
in dollars. Jan has purchased such a car, and decides to budget between $6400 and $7100
for next year’s driving costs. What is the corresponding range of miles that she can drive
her new car?
12. Use the graph to answer the following questions.
a) For what number of units does C(x) = R(x)?
b) On what interval is C(x) < R(x) ?
c) What is the break even point?
d) What is the interval where a profit is made?
e) Give the equation for each function.
13. A company’s daily profit (in dollars) from the production and sale of electrical components
can be described by the equation P( x)  3.75 x  1005 , where x is the number of units
produced and sold. How many units must they sell in order to avoid having a loss?
For each of the given quadratic functions, tell a) whether the parabola is concave up or
down, b) whether the vertex is a maximum or a minimum, and c) what the vertex is.
14. y  52 ( x  1) 2  3
15. y  2( x  8) 2  8
16. y  3x 2  18 x  3
Find the x-intercept(s), if any, the y-intercept, and the vertex of each quadratic function.
Then use those points (and others if necessary) to graph the function.
17. y  x 2  8x  7
18. y  6 x  x 2
19. y  ( x  2) 2  3
20. The profit for a product can be described by the function P( x)  40 x  3000  0.01x 2
dollars, where x is the number of units produced and sold. To maximize profit, how many
units must be produced and sold? What is the maximum possible profit?
21. The annual total cost for a product is given by C ( x)  5000  16 x  0.04 x 2 dollars, where
x is the number of units produced. To minimize the annual cost, how many units should be
produced? What is the minimum possible annual cost?
Solve each equation by factoring.
22. x 2  x  6  0
23. 10 x 2  30 x  0
24. 25 x 2  9  0
Solve each equation by the square root method.
25. 5 x 2  20
26. x 2  40  0
27. ( x  7) 2  9  0
Solve each equation by using the quadratic formula.
28. 3x 2  6 x  12  0
29. 2 x 2  5 x  7
30.  4 x 2  19 x  5  0
31. The total revenue for a product is given by R  266x , and the total cost for this same
product is C  2000  46 x  2 x 2 , where R and C are each measured in thousands of dollars
and x is the number of units produced and sold.
a) Write the profit function for this product from the two given functions.
b) What is the profit when 55 units are produced and sold?
c) How many units must be sold to break even on this product?
32. The demand for a product is given by p  7000  2 x dollars, and the supply for this product
is given by p  0.01x 2  2 x  1000 dollars, where x is the number of units demanded and
supplied when the price per unit is p dollars. Find the equilibrium quantity and price.
33. Given the function f ( x)  5x 0.2 , do the following.
a) Graph the function on the window [-10, 10] ×[-10, 10]. Sketch your graph.
b) Tell whether the function is concave up or down in quadrant I.
c) Find f (7) and f (4) . Round your answers to the nearest hundredth.
34. Given the piecewise-defined function below, find each of the following:
a) f (7)
b) f (3)
c) f (0)
d) f (4)
e) f (7)
 x 2  3 if x  0
f ( x)  
3 if x  0
35. The U.S. population (in thousands) can be modeled by the function f ( x)  165.6 x1.345 ,
where x is the number of years after 1800.
a) What type of function is this?
b) Find f (50) and give its meaning in this context.
c) What does the model predict the population will be in 2050?
36. The function f ( x)  92.239 x 0.1585 gives the percent of voter turnout during presidential
election years, with x representing the number of years after 1950.
a) Graph the function on a window which represents the years from 1950 to 2008, showing
percents from 0 to 100. Sketch your graph.
b) Does this model indicate that the percent of voter turnout is increasing or decreasing?
c) What does the model predict the percent of voter turnout to be in 2008? (Round to the
nearest whole percent.)