# Math

1. If the profit from the sale of x units of a product is P = 105x − 300 − x

2

, 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 + x

2

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 = 4.61×2 + 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 − 0.2×2 − 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 − 0.7×2. (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?