Chapter 2 Review Questions
Click on the question number to get to the solution.
[1] Ocean Marina rents moorage space for boats. Its charge for boats between 20 and 40 feet long is $180 plus $12 a foot.
Let x = boat length in feet C = cost of moorage:
x | 20 | 24 | 28 | 32 | 36 | 40 |
C |
- Complete the above table relating length and cost.
- State the equation describing cost as a function of length.
- Graph cost as a function of length for 20 to 40 feet.
[2] An agent has available truck shipping capacity for 62.5 tons of materials A and B. Each unit of A weighs 2.4 tons and each unit of B weighs 1.7 tons. Find an equation that relates the amounts of A and B that can be shipped.
[3] A company finds that when the temperature is 12°C it uses 100 litres of heating oil per day. When the temperature is 5°C, it uses 170 litres of heating oil per day.
- Mark the above data points on a graph (allow for negative temperatures).
- Find the equation (assumed to be linear) for oil usage in terms of temperature.
- Find the amount of oil that would be used at -5°C. Check your result on a graph.
[4] A company has a choice between two copiers, A and B. A costs $120 a month plus $0.05 per page; B costs $250 a month plus $0.03 per page.
- Find the cost equations for each copier.
- Which copier would be best for 5,000 pages a month? For 10,000 copies?
- Graph the equations on the same axes and show clearly where each copier is cheapest.
- Find the volumes at which the costs are equal and mark the point on your graph.
[5] Jones Stereo Company sells stereo sets for $150 each. The parts for each stereo cost $39.50 and the labor costs $53 per set. Fixed costs are $16,000 a month.
- Find Jones’ cost and revenue functions.
- How many must Jones produce and sell every month in order to (i) break even? (ii) make a $6,000 profit?
[6] Find the equations of the following lines.
- Passing through (3, 10) and (I, 6).
- With slope 1.5 and passing through (2, 5).
- Passing through (1, 7) and falling 2 units in for each increase of 1 in x.
[7] Jimms Company believes that the time required to produce its widgets is a linear function of the number of widgets to be produced in a run. It finds that to produce a run of 600 widgets requires 7,050 minutes, and that to produce a run of 350 widgets requires 4,300 minutes.
- Find the equation giving time required as a function of the number of widgets.
- How long will it take to produce 500 widgets?
- How many widgets can one expect to produce with a run that is assigned 81 hours?
[8] Deluxe Tire Company finds that to produce each additional tire it costs $21.50. The fixed costs of the plant operation amount to $126,000 per month. The tires are sold for $58 each.
- Find the revenue and cost functions for the operation.
- How many tires must be produced and sold in order for Deluxe to break even in a month?
- How many tires must be produced and sold in order for Deluxe to earn a profit of $40,000 in a month?
[9] A company can rent a car using either of two methods of payment. Method A requires $20 per day and $0.20 per kilometer. Method B requires $35 per day and $0.12 per kilometer.
- Which method is cheaper for 300 kilometres of use in one day?
- At what level of use per day are the costing methods equally expensive?
- A competitor to the company charged $37 for a day’s use during which 80 km were driven, and $43 for a day during which 120 km were driven. Assume the cost function to be linear and find the cost, C, as a function of distance traveled, x.
[10] Solve the following system of equations for x and y:
[latex]\begin{align*} y = 2.0 + 5.2x\\ y = 3.0 - 2.0x\\ \end{align*}[/latex]
[11] Solve the following system of equations for x and y:
[latex]\begin{align*} y-4x = 6\\ 2x + 3 y= 4\\ \end{align*}[/latex]
[12] A publishing company finds that, when it prices its computer books at $20 per book, it can sell 7,000 books per month. When it prices them at $25 per book, it can sell 5,500 books per month. The company assumes the relationship is linear.
- Find the equation which gives the number of books that can be sold per month in terms of the price.
- How many books should it be able to sell at $22?
[13] FG Company produces housings for a major electrical manufacturer. In a four-week period, during which it produced 2,550 housings, its total costs were $120,000, and in a four-week period, during which it produced 1,750 housings, its total costs were $98,000.
Assume that the costs are linear functions of the number of housings produced each four-week period.
- Find the equation relating total costs to the number of housings produced in a four-week period.
- According to your equation in (a), what are the following?
- the fixed costs per four-week period
- the variable cost per housing
- If the housings are sold for $55 each, how many must be produced and sold each four-week period in order to break even?
- Graph your results for costs and revenues. Identify all major areas on the graph.
[14] A furniture company finds that to make a run of 6,200 cabinets costs $205,000 and run of 8,300 costs $267,000. Assume the relationship between cost and number of cabinets produced is linear.
- Find the equation which gives cost as a function of number of cabinets produced.
- If cabinets are sold for $38 each, what is the minimum number of cabinets in a run such that the sales revenue would pay for the cost of the run?
[15] For a certain good there is a demand for 8,000 units when the price is $8/unit and a demand for 6,200 when the price is $12/unit. Assuming demand, d , is a linear function of price, p, find the slope of the line and the equation.
[16] HJ Outdoor Company found that it cost $6,100 to make a run of 105 jackets, and $7,900 to make a run of 150 jackets. Assume the cost is a linear function of the number of jackets made.
- Find the cost as a function of the number of jackets made in a run, and graph it from 80 to 200 jackets.
- Estimate the cost of a run of 180 jackets.
- If HJ sells the jackets for $55 each, how many must be in a run in order to barely recover the cost of making the jackets?
[17] Find the equation of the line:
- with slope 3 that contains the point (4, 1).
- with slope -5 that contains the point (-2, 3).
- containing the points (2, 3) and (-6, 1).
- containing the points (12, 16) and (1, 5).
- containing the points (-4, 5) and (-2, -3).
- that contains the point (-4, 2) and falls 2 units in y for every one unit increase in x.
[18] You use your calling card to make a telephone call to your friend who lives in Oyster River. You receive your monthly telephone bill and two of the items are as follows:
Time of call | Area | Amount |
5 min | Oyster River | $2.50 |
11 min | Oyster River | $4.60 |
- Define the two variables and create an equation to calculate the total cost of a call.
- How much would a 60-minute call cost?
[19] You have been asked to predict the cost of flying a Dash 7 aircraft from Vancouver to Seattle. You are provided with the following information:
Number of Passengers | Cost |
12 | $7,680 |
15 | $7,725 |
- Define the two variables and create an equation to calculate the total cost of a flight.
- How much would it cost to fly the plane empty (with no passengers)?
- Interpret the y intercept using the words of the problem.
- How much extra does it cost to have one more passenger?
- Interpret the slope using the words of the problem.
- How much would it cost to fly 14 passengers?
[20] B. Furniture Co. believes that the cost to produce its chairs is a linear function (straight line relationship) of the number of chairs to be produced in a run. It finds that to produce a run of 130 chairs requires $8,200, and that a run of 250 chairs requires $13,000.
- Find the equation giving cost, C, based on the number of chairs produced, x, in a run.
- What is the cost if no chairs are produced?
- Interpret the y intercept using the words of the problem.
- What is the extra cost to make one more chair?
- Interpret the slope using the words of the problem.
- How much would it cost to produce 400 chairs?
- How many chairs can be expected to be produced with a run that is assigned $8,600?
[21] Mr. Smith wants to rent a truck for one day to make a number of deliveries. He can rent the type of truck he needs from either of two companies. Company A charges $100 plus $0.40 per kilometre. Company B charges $40 plus $0.80 per kilometre.
- Write equations for the cost, C, of each truck in terms of x, the number of kilometres traveled.
- Suppose you rent a truck from Company A,
- How much would it cost to rent the truck but not actually drive it (0 km)?
- How much would it cost to drive 300 kilometres?
- Suppose you rent a truck from Company B,
- How much would it cost to rent the truck but not actually drive it (0 km)?
- How much would it cost to drive 300 kilometres?
- Graph the equations on the same axes and show on the graph where each company’s deal is best. Graph from 0 to 300 km. Use a large scale, graph paper and straight edge, and apply proper conventions.
- Using your graph, determine the point where the costs are equal. Mark the point on the graph you made in (d).
[22] Best Buy Furniture Store manufactures and sells bedroom suites. Each suite costs $800 and sells for $1,500. Fixed costs total $150,000.
- Write down the cost equation and the revenue equation.
- Find the breakeven point in units (number of suites).
[23] A company has determined that a minimum of 25,000 units of their product can be sold at a selling price of $10 per unit. However, if the selling price was reduced to $8.00 per unit, a minimum of 35,000 units can be sold. Relevant cost data for this product is as follows:
Fixed Costs | Variable Costs | |
$75,000.00 | Labour cost per unit | $4.50 |
Material cost per unit | $1.50 |
- Determine the breakeven points in units.
- Determine the profit earned by selling the minimum quantity for each price alternative (i.e., at both $10 and $8).
[24] CK Air has begun a new discount air service to Port Alberni. It costs $2,070 to fly an empty aircraft with crew to Port Alberni. Each passenger costs an extra $12 in food and extra fuel costs. Tickets sell for $150 for a one-way flight.
- Write down the cost equation and the revenue equation.
- How many passengers must the plane hold for the company to break even?
- Calculate the profit if 17 passengers fly to Port Alberni.
- If the company wants to make a profit of $690 per trip, how many seats must they sell?
- What is the contribution margin?
[25] Finicky-Cat Gourmet Pet Food makes organic cat food and sells them in 2 kg bags. The company has annual fixed costs of $150,000 and variable costs of $4 per bag. Finicky-Cat sells each bag for $10. Production capacity is 50,000 bags per year.
- Find the revenue and cost equations.
- How many bags of cat food does the company have to sell to break even?
- What are total sales at the breakeven point?
- What is the percent capacity at the breakeven point?
- The company anticipates that it will be able to make and sell 40,000 bags of cat food this year. What will it cost to produce these 40,000 bags?
- Graph the cost equation and the revenue equation on graph paper. Graph from 0 to 50,000 bags. Make sure to label the axes.
- Clearly identify the breakeven point on the graph.
- Identify the area on the graph where the company makes a profit and where it has a loss.
- Finicky-Cat wants to increase its selling price from the current $10 so that it could make a profit of $150,000 from selling 40,000 bags of cat food. What price must they charge?
[26] It costs Brawn Products $8 to make a particular model of shaver. The fixed costs are $12,000 per month. The shavers are sold to retailers for $25 less trade discounts of 33.5%, 10%.
- Write down the revenue and cost equations.
- Compute the breakeven point (in units and in sales dollars).
- Find the profit if 2,200 shavers are sold in a month.
- Find the profit if monthly sales (in dollars) are $37,500.
- Brawn wants to make a profit of $5,000/month. How many shavers must be sold?
- If the fixed cost increased to $18,000/month, how many shavers must they sell to break even?
- If the fixed costs remain the same (at $12,000/month) but the selling price is reduced to $10, what would the new breakeven point be?
- Brawn Products wants to add a third discount to reduce their selling price to $10 (as in part g). Find the rate of the third discount.
- The cost to make a shaver has increased by 25% from the current $8 and fixed costs have fallen by 25% from the current $12,000/month. If Brawn wants to make a profit of $20,000/month from the sale of 5,000 shavers, what should the new selling price be?
[27] Your company needs to lease a photocopier. It has a choice between three photocopiers A, B, and C. Copier A charges a flat fee of $3,000 per month. Copier B charges $2,000 per month plus $0.05 per copy; and copier C charges $1,000 per month plus $0.15 per copy.
- Find the cost equation for each copier.
- If the requirement is 15,000 copies per month which copier is cheapest?
- Calculate the points of indifference based on cost (i.e., determine the number of photocopies which will make the costs equal).
- between copier A and B
- between copier A and C.
- between copier B and C.
- On the same graph, graph the three cost equations over the range 0-30,000 copies.
- Over what range of values of x (the number of copies) is it most cost-effective (cheapest) to rent from A, B, or C?
-
a.
x 20 24 28 32 36 40 C 420 468 516 564 612 660 - 62.5=2.4A+1.7B ↵
- b. y = 220 −10x, where x is the temperature and y is oil usage; c. 270 litres ↵
- a. Copier A: C =$120 +$0.05x (x = number of pages; C = total cost); Copier B: C =$250 +$0.03x b. 5,000/month: Total cost for A= $370 Total cost for B = $400 Copier A is cheaper 10,000/ month: Total cost for A= $620 Total cost for B = $550 Copier B is cheaper d. Costs are equal when x = 6,500 ↵
- a. C =$16,000 +$92.50x (C = total costs; R = total revenue; x = number of stereos sold); R =$150x b. Breakeven = 279 stereos (rounded up from 278.26) For a $6,000 profit number of stereos= 383 ↵
- a. y= 4 + 2x b. y = 2 + 1.5x c. y = 9- 2x ↵
- a. y=450+11x (y = time in minutes; x = number of widgets) b. 5,950 minutes or 99.17 hours c. 401 widgets ↵
- a. R= 58x; C = $126,000 + 21.50x; x = number of tires b. 3452.05 round up to 3453 tires c. 4547.9 round up to 4548 tires ↵
- a. Method B is cheaper by $9.00 b. 187.5 kilometres c. C=$25+$0.15x ↵
- (0.1389, 2.7222) ↵
- (-1, 2) ↵
- a. b =$13,000 −$300p; (p = price; b = number of books) b. 6,400 books ↵
- a. C = $49,875 + $27.50h; (C = total costs; h = number of housings) b. fixed costs= $49,875; variable costs= $27.50 per housing c. 1,814 housings ↵
- a. C = $21,952 + $ 29.52x; (C = total cost; x = number of cabinets produced) b. 2,589 cabinets ↵
- d = 11,600 − 450p ↵
- a. C = $1,900 + $40x; (C = total cost; x = number of jackets made) b. $9,100 c. 127 jackets ↵
-
- y = −11 +3x
- y = −7−5x
- y = 2.5 + 0.25x
- y = 4+x
- y = −11 −4x
- y =−6−2x
-
- C =$0.75 +$ 0.35x where x = length of time of the call (in minutes); C = total cost of the call.
- $21.75
-
- C = $7,500 + $15x where x = number of passengers; C = total cost to fly the plane
- $7,500
- It would cost $7,500 to fly the plane empty (i.e., with no passengers).
- $15
- It costs an extra $15 for each additional passenger.
- $7,710
- a. C =$3,000 + $40x b. $3,000 c. The cost incurred even when no chairs are produced i.e., the overhead or expenses, (e.g., rent, property taxes). d. $40 e. Each additional chair made will cost $40 (the cost of labour and materials). f. $19,000 g. 140 chairs ↵
- a. CA=$100 +$0.40x, CB=$40+$0.80x
b. and c.
# Km x Total Cost CA Total Cost CB 0 $100 $40 300 $220 $280 - a. C = $150,000 + $800x; R = $1,500x; x = the number of bedroom suites produced and sold b. 215 suites (rounded up from 214.3 units) ↵
- a. 18,750 units/ 37,500 units b. $25,000 profit/ $5,000 loss ↵
-
- C =$2,070 +$12x; R =$150x ; x = the number of passengers on the plane
- 15 passengers
- $276 profit
- 20 passengers
- $138/passenger
- a. R =$10x; C =$150,000 +$4x; x = the number of bags produced and sold
b. 25,000 bags of cat food, revenue= $250,000; 50% capacity
c. $310,000
d.
Number of Bags Revenue Cost 0 0 $150,000 50,000 $500,000 $350,000 25,000 $250,000 $250,000 -
- R =$15x, C =$12,000 + $8x, x = the number of shavers produced and sold
- 1,715 shavers (rounded up from 1714.28). Sales: $25,725
- $3,400
- $5,500
- 2,429 shavers
- 2,572 shavers (after rounding up)
- 6,000 shavers
- 33.33%
- $15.80/shaver
- a. CA= $3,000; CB= $2,000 + $0.05x; CC= $1,000 +$0.15x; where x = the number of copies
b. A: $3000; B: $2,750; C: $3,250 so B is cheapest for 15,000 copies
c. CA=CB at 20,000 copies; CA= CC at 13,333 copies; CB = CC at 10,000 copies
d. Plot these points below. Your lines must cross at the indifference points you found in (c). If not, your graph is wrong.
Number of copies (x) Cost A Cost B Cost C 0 $3,000 $2,000 $1,000 30,000 $3,000 $3,500 $5,500
Markup in dollars for one item