MAT 111 - Pre-Calculus Name Class Worksheet - Word Problems Chapter 1 1. The cost of a Frigbox refrigerator is $950, and it depreciates $50 each year. The cost of a new Arctic Air refrigerator is $1200, and its value depreciates linearly to $900 in three years. (a) Write a formula for the value of the Frigbox refrigerator, F, in terms of its age, t, in years. (b) Write a formula for the value of the Arctic Air refrigerator, A, in terms of its age, t, in years. (c) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value? (d) If both refrigerators continue to depreciate at the same rates, what happens to the values of the refrigerators in 20 years time and what does this mean?
2. Straight-line Depreciation Book value is the value of an asset that a company uses to create its balance sheet. Some companies will depreciate their assets using straight-line depreciation so that the value of the asset declines by a fixed amount each year. The amount of the decline depends on the useful life that the company places on the asset. Suppose that a company just purchased a fleet of new cars for its sales force at a cost of $28,000 per car. The company chooses to depreciate each vehicle using the straight-line method over 7 years. (a) Make a table and graph showing the value of the equipment over the seven-year period. (b) Give a formula for value, v, as a function of time, t. (c) Explain the meaning of the slope of this problem. (d) Explain the meaning of the v-intercept and t-intercept of this problem.
3. An empty champagne bottle is tossed from a hot-air balloon. Its upward velocity is measured every second and recorded in the table below. t (sec) 1 2 3 4 5 v (ft/sec) 8 24 56 88 120 (a) Describe the motion of the bottle in words. What do negative values of v represent? (b) Find a formula for v in terms of t. (c) Explain the physical significance of the slope of your formula. (d) Find and explain the physical significance of the t-axis and v-axis intercepts.
4. At a price of $2.05 per gallon, the average weekly demand by consumers for gasoline is 43 gallons. If the price rises to $2.80 per gallon, the weekly demand drops to 28 gallons. (a) Find a formula for Q, the weekly quantity of gasoline demanded, as a function of p, the price per gallon, assuming that demand is linear. (b) What is the economic significance of the slope of your formula (explain the slope)? (c) What is the Q-intercept and the p-intercept for your formula? (d) What is the economic interpretation of each; the Q-intercept and p-intercept (explain the intercepts from part b)? (e) Are these interpretations realistic?
5. You want to choose one long-distance telephone company from the following options. Company A charges $0.37 per minute Company B charges $13.95 per month plus $0.22 per minute. Company C charges a fixed rate of $50.00 per month. Let Y A, Y B, Y C represent the monthly charges using Company A, B, and C respectively. Let x be the number of minutes per month spent on long distance calls. (a) Find formulas for Y A, Y B, and Y C as functions of x. (b) Graph the monthly charges for each company on the same set of axes. (c) When is the monthly charge for Company A and Company B the same? (d) Find the x-values for which company B is the cheapest.
6. A business consultant works 10 hours a day, 6 days a week. She divides her time between meetings with clients and meetings with co-workers. A client meeting requires 3 hours while a co-worker meeting requires 2 hours. Let x be the number of co-worker meetings the consultant holds during a given week. If y is the number of client meetings for which she has time remaining, then y is a function of x. Assume the function is linear and that meetings can be split up and continued on different days. (a) If she holds no client meetings in a week, how many co-worker meetings can be held? (b) If she holds no co-worker meetings in a week, how many client meetings can be held? (c) Find a formula for y in terms of x. (d) Explain the meaning of the slope in terms of this problem. (e) Explain the meaning of the x and y intercepts in terms of this problem.
ANSWERS - Class Worksheet - Word Problems Chapter 1 1. (a) F 950 50t (b) A 1200 100t (c) Can use graphing calculator and find point of intersection. F A or 950 50t 1200 100t t = 5 years (d) F 950 50 20 50 A 1200 100 20 800 In 20 years, the refrigerators will have NO value or will be worthless. A negative value is not possible. 2. (a) t (time in years) V (value in thousands of dollars) 0 28 1 24 2 20 3 16 4 12 5 8 6 4 7 0 (b) V 28,000 4000t (c) 4000 value m which means that each year the car s value depreciates by $4,000 1 year (d) v-intercept is (0, 28000) - The car s initial value is $28,000 t-intercept is (7, 0) - In 7 years, the car s value will be $0
3. t (sec) 1 2 3 4 5 v (ft/sec) 8 24 56 88 120 (a) The bottle will go up in the first second and then start to come down between 1 and 2 seconds. Negative v values represent the velocity of the bottle as it is going down or it represents downward velocity. (b) Use any two points from the table and find slope. 24 8 With (1, 8) and (2, 24), m 32 2 1 m 32 and (1, 8) v b mt 8 b 32 1 So, v 40 32t (c) m 32 velocity in ft per second 1 time in seconds This means that each second (1 second), the champagne bottle s velocity decreases by 32 feet per second. (d) t - intercept, (1.25, 0), is found by substituting v = 0 and solving for t After 1.25 seconds, the velocity is zero which represents when the bottle stops and then will begin to come down. v - intercept, (0, 40) is b, or the initial upward velocity of the champagne bottle is 40 feet per second when it is first tossed from the hot air balloon. 4. (a) Q f p 84 20 p (b) For an increase in price of $1.00 per gallon, there will be a 20 gallon decrease in weekly demand. (c) 0, 84 is the Q-intercept 4.20, 0 is the p-intercept (d) The Q-intercept which is the initial demand means that at a price of $0 dollars, there will be a weekly demand of 84 gallons of gasoline. The p-intercept which means that at a price of $4.20 dollars per gallon of gasoline, there will be NO weekly demand of gasoline. (e) Both intercepts are NOT realistic because the price of gas will never be $0 and there will always be some demand for gas.
5. (a) (b) Y Y Y A B C 0.37x 13.95 0.22x 50 (c) YA YB x 93 minutes (d) This is the x values when the Y B line is below (smaller outputs) the other two lines. 93 x 164 or company B will be the cheapest when the number of minutes used is between 93 to 164. 6. (a) 30 co-worker meetings can be held, which means (30, 0) (10 hours/day)(6 days/week) = 60 hours/week 2x = 60 so x = 30 (b) 20 client meetings can be held, which means (0, 20) 3y = 60 so y = 20 (c) 2x 3y 60 y 20 2 3 x (or use two points from parts (a) & (b) and find slope, then b) (d) 2 number of client meetings per week m 3 number of co-worker meetings per week For every three additional co-worker meetings per week, there will be two less client meetings per week. (e) y - intercept is (0, 20) which means that if she holds no co-worker meetings in a week then she can hold 20 client meetings that week. x - intercept is (30, 0) which means that if she holds no client meetings in a week then she can hold 30 co-worker meetings that week.