math
1. If the profit from the sale of x units of a product is P = 105x − 300 − x2, what
level(s) of production will yield a profit of $1050? (Enter your answers as a
comma-separated list.)
x = _________ units
2. The total costs for a company are given by C(x) = 5400 + 80x + x2
and the total revenues are given by
R(x) = 230x.
Find the break-even points. (Enter your answers as a comma-separated list.)
x= __________ units
3. If total costs are C(x) = 900 + 800x and total revenues are R(x) = 900x − x2, find the break-even points. (Enter your answers as a comma-separated list.)
x= _____________
4. For the years since 2001, the percent p of high school seniors who have tried marijuana
can be considered as a function of time t according to
p = f(t) = 0.17t2 − 2.61t + 52.64
where t is the number of years past 2000.† In what year after 2000 is the percent
predicted to reach 75%, if this function remains valid?
_______________
5. Using data from 2002 and with projections to 2024, total annual expenditures for
national health care (in billions of dollars) can be described by
E = 4.61x2 + 43.4x + 1620
where x is the number of years past 2000.† If the pattern indicated by the model
remains valid, in what year does the model predict these expenditures will reach
$15,315 billion?
__________________
6. The monthly profit from the sale of a product is given by P = 32x − 0.2x2 − 150 dollars.
(a) What level of production maximizes profit?
___________ units
(b) What is the maximum possible profit?
$_____________
7. Consider the following equation.
y = 9 + 6x − x2
(a) Find the vertex of the graph of the equation.
(x, y) = (__________)
(b) Determine what value of x gives the optimal value of the function.
x=_____________
(c) Determine the optimal (maximum or minimum) value of the function.
y=______________
8. Consider the following equation.
f(x) = 6x − x2
(a) Find the vertex of the graph of the equation.
(x, y) = (__________)
(b) Determine what value of x gives the optimal value of the function.
x=_____________
(c) Determine the optimal (maximum or minimum) value of the function.
f(x)= _____________
9. Find the maximum revenue for the revenue function R(x) = 358x − 0.7x2. (Round your
answer to the nearest cent.)
R = $______________
10. The profit function for a certain commodity is P(x) = 150x − x2 − 1000. Find the level of
production that yields maximum profit, and find the maximum profit.
x= _________ units
P=$ _________
11. If, in a monopoly market, the demand for a product is p = 2000 − x and the revenue is
R = px, where x is the number of units sold, what price will maximize revenue?
$________________
12. If the supply function for a commodity is p = q2 + 6q + 16 and the demand function is p
= −3q2 + 4q + 436, find the equilibrium quantity and equilibrium price.
equilibrium quantity_______________
equilibrium price $_______________
13. If the supply and demand functions for a commodity are given by p − q = 10 and q(2p
− 10) = 3600, what is the equilibrium price and what is the corresponding number of
units supplied and demanded?
equilibrium price $_______________
number of units _________ units
14. Suppose the percent of the total work force that is female is given by
(a) From the equation, identify the maximum point on the graph of y = p(t). (Round
your answers to two decimal places.)
(t, y) = (____________)
(b) In what year is the percent of women workers at its maximum, according to this
model?