Problems with (*) are important

Problems with (**) are harder or higher level

Problems with (B) are bonuses

PART (A): Cost, Revenue & Profit

1) A small home business is set up with an investment of $10,000 for equipment. The business manufactures a product at a cost of $0.65 per unit. If the product sells for $1.20, how many units must be sold before the business breaks even?
 
2) A person setting up a part-time business makes an initial investment of $5,000. The unit cost of the product is $21.60, and the selling price is $34.10.
a) Find equations for the total cost C and total revenues R for x units.
b) Find the break-even point.
3) A manufacturing company determines the total cost in dollars of producing x units of a certain product is $25 per unit plus a fixed cost of $3500. Find the equation of the cost, graph it, find the y-intercept and the slope.
 
*4) In a manufacturer, the cost of producing 20 flashlights per week is $200 while the cost of producing 100 flashlights per week is $600.
a) Find the equation for the cost.
b) Find the cost of producing 150 flashlights per week.
c) What is the manufacturer fixed cost?
 
*5) A contractor purchases a piece of equipment for $26,500. The equipment requires an average expenditure of $5.25 per hour for fuel and maintenance, and the operator is paid $9.50 per hour.
a) Find a linear equation giving the total cost C of operating this equipment t hours.
b) Given that customers are charged $25 per hour of machine use, write an equation for the revenue R derived from t hours of use.
c) Use the formula (P=R-C) to write an equation for the profit derived from t hours of use.
d) Find the number of hours the equipment must be operated for the contractor to break even.
 
6) Suppose the cost for a product is $2.50 per unit with a fixed cost of $1200. If the selling price per unit is $10, how many units must be produced to break even? (graph the equations and show the intersection point)
 
B7) Use the data in the table below to solve the following problem. (R-value of insulation is a measure of its ability to resist heat transfer, higher R-value means higher insulation)
a) Write linear equations relating the R-value to the thickness in inches for fiberglass insulation and cellulose fiber.
b) Use the costs per cubic inch to determine which product costs less and provides an R-value of 24.
  Thickness (in) R-value Cost/cubic inch $
Fiberglass 3.5 11 0.21
  6 19  
Cellulose 3.5 13 0.33
  6 22  
 
8) A manufacturer sells belts for $12 per unit. The fixed costs are $1600 per month, and the variable costs are $ 8 per unit. a) Write the equations of the revenue and the cost function. b) Find the break-even point.
 
*9) The total cost per week of producing a calculator is C = 360 + 10x + 0.2x2. If the price per unit sold is 50 - 0.2x, at what level of production will the break even point(s) occur? graph the problem.
 
10) A manufacturer determines that the revenue generated by selling x units of a product is a linear function of x. If the revenue from 20 units is $380 and the revenue from 15 units is $285, find the revenue function.
 
11) The Green-Belt Company determines that the cost of manufacturing men's belts is $2 each plus $300 per day in fixed costs. The company sells the belts for $3 each. What is the break-even point? graph the problem.
 
*12) Scott wants to buy a total of 500 shares in the stock market. He will buy x shares at $4 per share and the rest at $6 per share. Write the equation for the total cost of the shares. What will be Scott's cost if he buys 200 shares at $4 ?
 
*13) The manager of a pie shop sells his pies for $6.5. The overhead is $378 per day and each pie costs $1.10 to make.
a) Write the revenue function, the cost function and the profit function.
b) Write the marginal revenue, cost and profit function.
c) Find the break even point.
 
14) A manufacturer of golf clubs finds that the fixed costs are $5780 per week and the cost of producing each set is $73.00. Each set of clubs can be sold for $243.00.
a) Write the revenue function and the cost function.
b) Write the profit function and the marginal profit function.
c) Find the break even point.
 
15) A grocery store bought ice cream for 59 cents per quart and stored them in two freezers. During the night, one freezer "defrosted" and ruined 14 quarts. If the remaining ice cream was sold for 98 cents per quart, find the function for the profit in terms of the number of quart bought.
 
*16) The quantity sold x of a certain kind of radio is inversely proportional to the price p. It was found that 240,000 radios will be sold when the price per radio is $12.50. How many will be sold if the price is $18.75?
 
*17) The time t required to do a certain job varies inversely as the number of people P who work on the job. It takes 4 hours for 12 people to erect some football bleacher. Find the variation constant and how long would it take 3 people to do the same job?
 
*18) The amount of garbage G produced in the United states varies directly as the number of people N who produce the garbage. It is known that 50 tons of garbage is produced by 200 people in 1 year. The population in San Francisco is 705,000. Find the variation constant and the amount of garbage produced in San Francisco in 1 year.
 
19) A bicycle manufacturer has a fixed cost of $500 per week and a variable cost of $30 per bicycle. The manufacturer sells the bicycles for $80 each.
a) Find the cost function C(x), the revenue function R(x).
b) Find the profit function P(x).
c) Find the output levels x at which P(x) is positive . Graph the functions.
 
*20) A lawnmower manufacturer sells mowers to a certain retailer for $200 each, plus a handling charge of $75 on each order. The retailer applies a markup of 40% to the total price paid for the manufacturer.
a) Find the function C giving the retailer's total cost of a single order x lawnmowers.
b) Find the function R giving the retailer's total revenues from the sale.
c) If the retailer sells all these lawnmower at the same price, what is the retail price per lawnmower?
ANSWERS:
  1. 18,182 units
  2. a) C = 21.6x + 5000, R = 34.1x,
    b) 400 units
  3. C=25x + 3500, y-intercept = 3500 , slope = 25
  4. a) C(x)= 5x + 100,
    b) $850,
    c) $100 per week
  5. a) C = 14.75t + 26,500
    b) R=25t
    c) P = 10.25t - 26,500
    d) 2585.4 hours
  6. 160 units
  7. a) R=3.2t-0.2, R = 3.6t + 0.4
    b) fiberglass $ 1.58, cellulose $ 2.16
  8. a) R = 12x; C = 8x + 1600
    b) 400 units
  9. at x = 10 and at x = 90
  10. R = 19 x
  11. x = 300 belts.
  12. C = 3000 - 2x, $ 2600.
  13. a) R = 6.50x, C = 1.10x + 378. P = 5.40x -378
    b) 6.50; 1.10; 5.40 .
    c) x = 70 pies.
  14. a) R = 243x, C = 73x + 5780
    b) P = 170x - 5780; marginal prof. 170
    c) x = 34 sets.
  15. P(x) = 0.39x - 13.72
  16. 160,000
  17. k = 48, t=16 hours.
  18. k = 0.25, G = 176,250 tons
  19. a) C(x)= 500 + 30x, R(x) = 80x,
    b) P(x) = 50x - 500,
    c) x > 10
  20. a) C(x)=200x + 75,
    b) R(x)=280x + 105,
    c) Pr(x)=280 + 105/x