Chapter 2 Review Questions

^[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:

1. Complete the above table relating length and cost.
2. State the equation describing cost as a function of length.
3. 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.

1. Mark the above data points on a graph (allow for negative temperatures).
2. Find the equation (assumed to be linear) for oil usage in terms of temperature.
3. 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.

1. Find the cost equations for each copier.
2. Which copier would be best for 5,000 pages a month? For 10,000 copies?
3. Graph the equations on the same axes and show clearly where each copier is cheapest.
4. 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.

1. Find Jones’ cost and revenue functions.
2. 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.

1. Passing through (3, 10) and (I, 6).
2. With slope 1.5 and passing through (2, 5).
3. 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.

1. Find the equation giving time required as a function of the number of widgets.
2. How long will it take to produce 500 widgets?
3. 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.

1. Find the revenue and cost functions for the operation.
2. How many tires must be produced and sold in order for Deluxe to break even in a month?
3. 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.

1. Which method is cheaper for 300 kilometres of use in one day?
2. At what level of use per day are the costing methods equally expensive?
3. 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.

1. Find the equation which gives the number of books that can be sold per month in terms of the price.
2. 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.

1. Find the equation relating total costs to the number of housings produced in a four-week period.
2. According to your equation in (a), what are the following?
   1. the fixed costs per four-week period
   2. the variable cost per housing
3. If the housings are sold for $55 each, how many must be produced and sold each four-week period in order to break even?
4. 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.

1. Find the equation which gives cost as a function of number of cabinets produced.
2. 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.

1. Find the cost as a function of the number of jackets made in a run, and graph it from 80 to 200 jackets.
2. Estimate the cost of a run of 180 jackets.
3. 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:

1. with slope 3 that contains the point (4, 1).
2. with slope -5 that contains the point (-2, 3).
3. containing the points (2, 3) and (-6, 1).
4. containing the points (12, 16) and (1, 5).
5. containing the points (-4, 5) and (-2, -3).
6. 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:

1. Define the two variables and create an equation to calculate the total cost of a call.
2. 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:

1. Define the two variables and create an equation to calculate the total cost of a flight.
2. How much would it cost to fly the plane empty (with no passengers)?
3. Interpret the y intercept using the words of the problem.
4. How much extra does it cost to have one more passenger?
5. Interpret the slope using the words of the problem.
6. 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.

1. Find the equation giving cost, C, based on the number of chairs produced, x, in a run.
2. What is the cost if no chairs are produced?
3. Interpret the y intercept using the words of the problem.
4. What is the extra cost to make one more chair?
5. Interpret the slope using the words of the problem.
6. How much would it cost to produce 400 chairs?
7. 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.

1. Write equations for the cost, C, of each truck in terms of x, the number of kilometres traveled.
2. Suppose you rent a truck from Company A,
   1. How much would it cost to rent the truck but not actually drive it (0 km)?
   2. How much would it cost to drive 300 kilometres?
3. Suppose you rent a truck from Company B,
   1. How much would it cost to rent the truck but not actually drive it (0 km)?
   2. How much would it cost to drive 300 kilometres?
4. 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.
5. 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.

1. Write down the cost equation and the revenue equation.
2. 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:

1. Determine the breakeven points in units.
2. 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.

1. Write down the cost equation and the revenue equation.
2. How many passengers must the plane hold for the company to break even?
3. Calculate the profit if 17 passengers fly to Port Alberni.
4. If the company wants to make a profit of $690 per trip, how many seats must they sell?
5. 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.

1. Find the revenue and cost equations.
2. How many bags of cat food does the company have to sell to break even?
   1. What are total sales at the breakeven point?
   2. What is the percent capacity at the breakeven point?
3. 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?
4. Graph the cost equation and the revenue equation on graph paper. Graph from 0 to 50,000 bags. Make sure to label the axes.
   1. Clearly identify the breakeven point on the graph.
   2. Identify the area on the graph where the company makes a profit and where it has a loss.
5. 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%.

1. Write down the revenue and cost equations.
2. Compute the breakeven point (in units and in sales dollars).
3. Find the profit if 2,200 shavers are sold in a month.
4. Find the profit if monthly sales (in dollars) are $37,500.
5. Brawn wants to make a profit of $5,000/month. How many shavers must be sold?
6. If the fixed cost increased to $18,000/month, how many shavers must they sell to break even?
7. 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?
8. 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.
9. 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.

1. Find the cost equation for each copier.
2. If the requirement is 15,000 copies per month which copier is cheapest?
3. Calculate the points of indifference based on cost (i.e., determine the number of photocopies which will make the costs equal).
   1. between copier A and B
   2. between copier A and C.
   3. between copier B and C.
4. On the same graph, graph the three cost equations over the range 0-30,000 copies.
5. Over what range of values of x (the number of copies) is it most cost-effective (cheapest) to rent from A, B, or C?