Math Final Assignment

1. In a retailing operation, total costs can be expressed as a function of total sales. A small store found recently that when their total sales increased form $1,000 to $4,000, their costs increased from $750 to $1,500.
   1. Which is the dependent variable and which is the independent variable? Explain.
   2. Assuming that a linear functional relationship exists, determine the equation that relates total costs to total sales.
   3. As long as the total revenue is between $0 and $5,000, this relationship is expected to hold. Graph the function over the appropriate range.
   4. Explain why the nature of the relationship may change if sales exceed $5,000.
   5. Find and interpret C(2,000).
   6. Calculate the profit when sales are $2,000.
   7. What level of sales would be required in order for the company to achieve a profit of $2,500.
2. If the relationship between the total cost and the number of units produced is linear and if the total cost increases by $3 for each additional unit made:
   1. Find the equation of the total cost function C(x) using the fact that C(1,000) = 30,000 (dollars) where x is the number of units produced.
   2. Graph C(x) for 0 ≤ X ≤ 3000.
   3. Find and interpret each of the following:
      1. C(0)
      2. C(100)

1. C(1,001) - C(1,000)

1. The XYZ Company can sell 10,000 units of steel for $500,000. The cost of these units is $600,000. The company can sell 25,000 units for $1,250,000 and these cost $1,050,000. Assuming there is a linear relationship between these variables:
   1. Find the revenue, cost and profit functions.
   2. Find the breakeven point and graph the revenue and cost functions for 0 ≤ X ≤ 30000.
2. An auto parts company requires a large number of gaskets which they currently buy for .50 each. A recent feasibility study has indicated that if they produced them internally, their annual fixed costs (loan payments on equipment, equipment maintenance, etc.) would total $10,000 and the material and labour costs would be .40 per gasket.
   1. They currently require 70,000 gaskets per year. Should they begin to produce their own gaskets? Explain.
   2. They estimate their gasket requirements will increase by 10,000 a year. In how much time will it become profitable to manufacture the gaskets internally assuming that the costs remain the same.
3. The total monthly cost C(q) of a retail operation is linearly related to the quantity q, by the following equation: C(q) = 0.52 q + 30,000. They can sell each unit for $0.80.
   1. Interpret the two parameters of the cost equation.
   2. What is the profit equation P(q)?
   3. How many units must be sold to make a profit of $250,000?
4. If total cost, T(X), is related to output, X, by the equation:
   T(X) = 0.001X ² + 0.5X + 50
   1. What is the variable cost function?
   2. What is the average cost function?

If output is 100 units, find

1. Total fixed cost;
2. Total variable cost;
3. Total cost;
4. Average cost;
5. Marginal cost of the 100th unit.

1. A firm can sell X units of a product at a price of P cents per unit, where P = 503 - X
   The total costs of production is given by the function T(X) = 500 + 2X
   1. Find the function that relates total revenue to the number of units sold - R(X).
   2. Find the function that relates total profit to the number of units sold - P(X).
   3. Find the number of units to be sold to achieve maximum profit and find the maximum profit.
   4. What price per unit would the firm be charging at the maximum profit output level?
2. A Vancouver travel agency advertises all-expenses-paid trips to the Grey Cup Game for special groups. Transportation is by bus which seats 48 passengers, and the charge per person is $80, plus an additional $2 for each empty seat.
   1. If there are X empty seats, find the equations for:
      1. How many passengers are on the bus and
      2. How much does each pay?
   2. What are the travel agency's total revenue when there are X empty seats?
   3. Find the maximum revenue.
3. A quadratic revenue function often occurs in practice. The reason is that as more units are produced, the price must be lowered in order to sell them all.
   1. A company has found that if they produce x units of a product, they must sell them each at $10 - 0.2x if they are to sell all of them. Determine the revenue function.
   2. Their fixed costs are $40 and their variable costs per unit are $1. Determine the cost and profit functions.
4. A small corner store owner decides to advertise on local radio in order to help stimulate his sales. Radio station CFAL charges him a flat rate of $99, plus $20 per day for advertising. From the onset of the advertising the revenue increase can be described as (120 - x) dollars per day where x is the number of days the ad is run. He plans to run the ad for only 60 consecutive days.
   1. How many days must the ad run until the increased revenue just covers the cost of the advertising?
   2. How many days should he let the ad run in order to maximize his profit from the ad campaign?
   3. What is the maximum profit?
5. 1. A manufacturer makes two products A and B, each of which is processed in two departments, production and finishing. Each A takes 3 hours to produce and six hours to finish, whereas each B takes 5 hours to produce and 2 hours to finish. How many units of A and B can be produced and finished if exactly 24 hours are available in each department and all hours must be used?
6. 2. A company makes three products X, Y and Z. Each product requires processing by three machines A, B and C. The time required to produce one unit of each product is shown below.

   PRODUCT	MACHINE A	MACHINE B	MACHINE C
   X	1	2	2
   Y	2	8	3
   Z	2	1	4

   
   The machines are available for 200, 525 and 350 hours each month. How many units of each product can be manufactured per month if all three machines are utilized to their full capacity?
7. 3. A manufacturer produces three types of radios: deluxe, standard and economy. Each radio uses three different types of transistors: P, Q and R. The deluxe radio uses 2 P's, 7 Q's and 1 R. The standard contains 2 P's, 3 Q's and 1 R, and the economy model requires 1 P, 2 Q's and 2 R's.
   
   How many radios of each type can be constructed if the total number of transistors (P's, Q's and R's) available are 2200, 3400 and 1400 respectively and all transistors must be used?
8. 4. The table below shows the number of hours required in each of two departments to make one unit of various products A, B and C. For example, product B requires 1 hour of time in department I and 3 hours in department II.

   	HOURS REQUIRED PER UNIT OF PRODUCT
   DEPARTMENT	A	B	C
   I	1	1	9
   II	1	3	7

   
   Find the number of units of A, B, and C which could be made if department I has 75 hours available and Department II has 65 hours available. It is necessary that all of the available hours be used.
9. 5. A medical supply company has 1150 worker-hours for production, maintenance, and inspection. Using this and other factors, the number of hours used for each operation, P, M, and I, respectively, is found by solving the following system of equations. Find P, M and I.
   P + M + I = 1150
   P = 4I - 100
   P = 6M + 50
10. 1. Maximize z = 16x + 8y  subject to:
    2x + y ≤ 30
    x + 2y ≤ 24
    x ≥ 0
    y  ≥ 0
11. 2. A company is planning to purchase and store two items, gadgets and widgets. Each gadget costs $2.00 and occupies 2 square meters of floor space; each widget costs $3.00 and occupies 1 square meter of floor space. $1,200 is available for purchasing these items and 800 square meters of floor space is available to store them. Each gadget contributes $3.00 to profit and each widget contributes $2.00 to profit. What combination of gadgets and widgets produces maximum profit?
12. 3. A manufacturer produces two items, bookcases and library tables. Each item requires processing in each of two departments. Department I has 40 hours available and department II has 36 hours available each week for production. To manufacture a bookcase requires 2 hours in department I and 4 hours in department II, while a library table requires 3 hours in department I and 2 hours in department II. Profits on the items are $6.00 for a bookcase and $7.00 for a library table. If all units produced can be sold, how many of each should be made in order to maximize profits?
13. 4. A manufacturer has a maximum of 240, 360, and 180 kilograms of wood, plastic and steel available. The company produces two products, A and B. Each unit of A requires 1, 3 and 2 kilograms of wood, plastic and steel respectively; each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively, and each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively. The profit per unit of A and B is $4.00 and $6.00 respectively. How many units of A and B should be manufactured in order to maximize profits? What would the maximum profit be?
14. 5. Minimize z = 3x + 6y subject to:
    4x + y ≥ 20
    x + y ≤ 20
    x + y ≥ 10
    x ≥ 0
    y ≥ 0
15. 6. A feed company is developing a feed supplement from two grains A and B. Each kilogram of A contains 0.3 grams of protein and 0.2 grams of carbohydrates; each kilogram of B contains 0.9 grams of protein and 0.1 grams of carbohydrates. There must be at least 27 grams of protein and at least 8 grams of carbohydrates. If each kilogram of A costs 30 cents and each kilogram of B costs 50 cents what combination of quantities of grain A and grain B will minimize the cost of a package of the new feed supplement?
16. Find the compound amount if $6,400 is invested for 2 years at 12% compounded monthly. What difference would compounding daily make in this example?
17. Deposits of $1,000, $1,100 and $680 were made into a savings account, the first two years ago, the second 18 months ago, the third 6 months ago. How much is in the account now if the interest on all deposits is 12% compounded semi-annually?
18. A deposit of $2,000 earns interest at a rate of 14% compounded quarterly. After two and a half years the interest rate changes to 13.5% compounded monthly. How much is in the account after six years?
19. Which is a better rate of interest, 16% compounded quarterly or 16 1/4% compounded semi-annually?
20. Billy Burnett's grandfather willed Billy $10,000 payable on his twenty-first birthday. What will be the value of the bequest when Billy reaches 17 years of age if money is worth 8% compounded semi-annually?
21. Suppose you had three different offers for your used car. One person will give you $1,000 right now, another offers $1,200 six months from now and the other offers $1,600 two years from now. If interest is 16% compounded quarterly, which offer is worth the most.
22. At 16% compounded quarterly, how long would it take for money to triple?
    
    
23. At what annual nominal rate of interest will $6,900 earn $6,400 interest in five years?
24. You want to have $1,000,000 in your bank account when you turn 65 years old. Today is your 20 ^th birthday. As a birthday present you received $27,000 and you want to invest this amount. At what annual interest rate must you achieve to realize this goal?

If not explicitly stated always assume payments are made at the end of the period.

1. The Derr-McGee Manufacturing Company plans to build a new $50,000 warehouse seven years from now. They plan to accumulate the $50,000 in an account before beginning construction. If money is worth 7% compounded annually, how much must each year?s deposit be in order to accumulate $50,000 at the end of the seventh year?
2. The Johnson family plans to purchase a new home five years from now. How much will they have accumulated in a savings account if they deposit $500 at the end of every six months for five years? The savings account earns interest at the rate of 5 % compounded semi-annually.
3. The XYZ Television Company offers a machine for $200 down and $25 per month for one year. If interest is charged at 18% compounded monthly, find the actual cash value of the television.
4. Sam Jones plans to retire at age 65. He wants to supplement his retirement by buying an annuity that will provide $2,400 each year for 10 years. If money is worth 8% compounded annually, how much will Jones have to pay for the annuity at age 65?
5. Marie Richard purchases a new sports car for $35,500. What is the monthly payment if Marie agrees to pay for the car in 48 months if the car dealer is offering an interest rate of 6.5% APR? Marie wishes to pay off the loan completely after one year of monthly payments. How much would this lump sum payment be?
6. A loan of $10,000 at 10% compounded annually is being amortized over 6 years. Under the following headings, work out the first four lines of the amortization schedule using annual payments and determine the outstanding principal after the fourth payment has been made.
   

1. Mr. Strupp expects to retire in 12 years. Beginning one month after his retirement, he would like to receive $500 per month for twenty years. How much must he deposit into a fund today to be able to do so if the rate of interest on the deposit is 12% compounded monthly?

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