Posted: March 3rd, 2022

A company charting its profits notices that the relationship between the number of units

1. A company charting its profits notices that the relationship between the number of units sold, x, and the profit, P, is linear. If 190 units sold results in $380 profit and 240 units sold results in $2980 profit, write the profit function for this company.

P = _________

Find the marginal profit.

$____________

2. A company distributes college logo sweatshirts and sells them for $45 each. The total cost function is linear, and the total cost for 90 sweatshirts is $3951, whereas the total cost for 260 sweatshirts is $5991.

(a) Write the equation for the revenue function R(x).

R(x) = ___________

(b) Write the equation for the total cost function C(x).

C(x) = ____________

(c) Find the break-even quantity.

x = __________ sweatshirts

3. Suppose a certain home improvement outlet knows that the monthly demand for framing studs is 2,500 when the price is $4.25 each but that the demand is 3,700 when the price is $3.89 each. Assuming that the demand function is linear, write its equation. Use p for price (in dollars) and q for quantity.

________________

4. It has been estimated that a certain stream can support 88,000 fish if it is pollution-free. It has further been estimated that for each ton of pollutants in the stream, 1500 fewer fish can be supported. Assuming that the relationship is linear, write the equation that gives the population of fish p in terms of the tons of pollutants x.

________________

5. An electric utility company determines the monthly bill for a residential customer by adding an energy charge of 7.34 cents per kilowatt-hour to its base charge of $18.39 per month. Write an equation for the monthly charge y in terms of x, the number of kilowatt-hours used. (Let y be measured in dollars.)

________________

6. Suppose that from 2020 to 2060, the number of females in the U.S. under the age of 18, in millions, can be modeled by

N = 0.140x + 38.4

where x is the number of years after 2020.

(a) Viewing N as a function of x, what is the slope m of the graph of this function?

m = _________

(b) What does the model predict the population of females under 18 (in millions) will be in 2040?

__________million.

7.

8. A concert promoter needs to make $79,600 from the sale of 1740 tickets. The promoter charges $40 for some tickets and $60 for the others. Let x represent the number of $40 tickets and y represent the number of $60 tickets.

(a) Write an equation that states that the sum of the tickets sold is 1740.

______________

(b) Write an expression for how much money is received from the sale of $40 tickets?

______________

(c) Write an expression for how much money is received from the sale of $60 tickets?

_________________

(d) Write an equation that states that the total amount received from the sale is $79,600

___________________

(e) Solve the equations simultaneously to find how many tickets of each type must be sold to yield the $79,600.

x= ____________

y= ____________

9. If the demand for a pair of shoes is given by 2p + 5q = 200 and the supply function for it is p − 2q = 10, compare the quantity demanded and the quantity supplied when the price is $90.

quantity demanded ___________ pairs of shoes

quantity supplied ____________ pairs of shoes

10. Find the market equilibrium point for the following demand and supply functions.

Demand: p = −4q + 312 Supply: p= 6q + 1

(q, p) = ( __________ )

11. Find the equilibrium point for the following supply and demand functions.

Demand: p = −4q + 220 Supply: p= 16q + 20

(q, p) = ( __________ )

12. Retailers will buy 45 Wi-Fi routers from a wholesaler if the price is $10 each but only 20 if the price is $85. The wholesaler will supply 56 routers at $46 each and 70 at $50 each. Assuming that the supply and demand functions are linear, find the market equilibrium point.

(q, p) = ( __________ )

13.

(b) How long is it until the building is completely depreciated (its value is zero)?

__________months

(c) The point (70, 225,000) lies on the graph. Explain what this means.

The point (70, 225,000) means that after ________ months the value of the building will be $ _____________ .