Homework on Linear Functions
Mathematics 120—Precalculus Dr. Peratt
1. A stalactite grows according to the formula L(t) = 17.75+2501 t,where L is the length of the stalactite in inches and t is the time, in years, since the stalactite was first measured.
(a) Identify the slope, giving units.
(b) Give a practical interpretation of the slope.
(c) Identify the vertical intercept, giving units.
(d) Give a practical interpretation of the vertical intercept.
2. A phone company charges according to the formula C(n) = 29.99 + 0.05n, where n is the number of minutes and C is the monthly charge, in dollars.
(a) Identify the slope, giving units.
(b) Give a practical interpretation of the slope.
(c) Identify the vertical intercept, giving units.
(d) Give a practical interpretation of the vertical intercept.
3. A college meal plan requires that you pay a membership fee; then, all of your meals are at a fixed price per meal.
(a) If 90 meals cost $1,005.00 and 140 meals cost $1,205.00, write a linear function that gives the cost of a meal plan, C, in terms of the number of meals, n.
(b) What is the cost per meal and what is the membership fee?
(c) Find the cost for 120 meals.
(d) Find the maximum number of meals you can purchase on a budget of $1,285.00.
4. The cost of a Frigbox refrigerator is $950.00, and it depreciates $50.00 per year. The cost of an Arctic Air refrigerator is $1,200.00 and it depreciates $100.00 per year.
(a) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value?
(b) if both refrigerators continue to depreciate at the same rates, what happens to the values of the refrigerators in 20 years time? What does this mean?
(b) Use the TI-84 to create graphs of each function for 0 ≤ x ≤ 500.
(c) Identify the slopes and intercepts of each function, giving units, and interpret their meanings.
(d) Use the graph in part (b) to determine the circumstances under which Company A will be the least expensive option. Repeat for Companies B and C, and explain why this makes sense.
6. In 2003, the number N of cases of SARS (Severe Acute Respiratory Syndrome) reported in Hong Kong was initially approximated by N = 78.9 + 30.1t, where t is the number of days since March 17. Interpret the meanings of the constants 78.9 and 30.1.
7. In each case, sketch the graph of a linear function which satisfies the given conditions.
(a) A quantity q begins at 3 and increases by 2 per day.
(b) A quantity q begins at 8 and decreases by 3 per day.
(c) It has a small negative intercept and a large positive slope.
(d) It has a large positive intercept and a small negative slope.
Homework on Linear Functions—Solutions
Mathematics 120—Precalculus Dr. Peratt
1. A stalactite grows according to the formula L(t) = 17.75+2501 t,where L is the length of the stalactite in inches and t is the time, in years, since the stalactite was first measured.
(a) Identify the slope, giving units.
Answer: The slope is 2501 inches per year.
(b) Give a practical interpretation of the slope.
Answer: This means that, for every year time advances, the stalactite will grow by 2501 of an inch.
(c) Identify the vertical intercept, giving units.
Answer: The vertical intercept is 17.75 inches.
(d) Give a practical interpretation of the vertical intercept.
Answer: This means that, when the stalactite’s length was first mea- sured, it was 17.75 inches long.
2. A phone company charges according to the formula C(n) = 29.99 + 0.05n, where n is the number of minutes and C is the monthly charge, in dollars.
(a) Identify the slope, giving units.
Answer: The slope is 0.05 dollars per minute.
(b) Give a practical interpretation of the slope.
Answer: This means that, for each additional minute used, the monthly cost will rise by 0.05 dollars.
(c) Identify the vertical intercept, giving units.
Answer: The vertical intercept is 29.99 dollars.
(d) Give a practical interpretation of the vertical intercept.
Answer: This means that you will pay $29.99 per month simply for having access to the phone service, regardless of whether you actually make any calls.
3. A college meal plan requires that you pay a membership fee; then, all of your meals are at a fixed price per meal.
(a) If 90 meals cost $1,005.00 and 140 meals cost $1,205.00, write a linear function
(b) What is the cost per meal and what is the membership fee?
Answer: The membership fee is $645 and the cost per meal is $4.
(c) Find the cost for 120 meals.
Answer: The cost is C = 4(120) + 645 = $1, 125.00.
(d) Find the maximum number of meals you can purchase on a budget of $1,285.00.
Answer: We want C = 1285, so we solve 1285 = 4n + 645 → n = 160 meals.
4. The cost of a Frigbox refrigerator is $950.00, and it depreciates $50.00 per year. The cost of an Arctic Air refrigerator is $1,200.00 and it depreciates $100.00 per year.
(a) If a Frigbox and an Arctic Air are bought at the same time, when do the two refrigerators have equal value?
Answer: The value of the Frigbox is given by V = 950 − 50n, where n is the number of years since it was purchased. Similarly, the value of the Arctic Air is given by V = 1200 − 100n. We want to know when their values are equal, so we solve 950 − 50n = 1200 − 100n → n = 5 years.
(b) if both refrigerators continue to depreciate at the same rates, what happens to the values of the refrigerators in 20 years time? What does this mean?
Answer: If n = 20, then the Frigbox is worth V = 950 − 50(20) = −50 dollars and the Arctic Air is worth V = 1200 − 100(20) = −800 dollars.
What this means is that they no longer have any retail value (and hence do not have to be claimed on taxes as property, for example, if you were a company).
5. You need to rent a car and so compare the charges of three different companies. Com- pany A charges 20 cents per mile plus $20.00 per day. Company B charges 10 center per mile plus $35.00 per day. Company C charges $70.00 per day (no charge per mile).
(a) Find formulas for the cost of driving cars rented, for a single day, from each company as a function of x, the number of miles driven.
Answer: Company A: C = 20 + 0.20x, Company B: C = 35 + 0.10x, and Company C: C = 70.
(b) Use the TI-84 to create graphs of each function for 0 ≤ x ≤ 500.
Answer: In the graph below,Company A is represented by thered line, Company B by the blue line, and Company Cby the green line.
(c) Identify the slopes and intercepts of each function, giving units, and interpret their meanings.
Answer: The slope of Company A’s function is 0.20 dollars per mile, representing the variable cost per mile, and the intercept is dollars, rep- resenting the fixed cost per day. Similarly for Company B’s function. For Company C’s function, the slope is 0, indicating that there is no variable cost per mile.
(d) Use the graph in part (b) to determine the circumstances under which Company A will be the least expensive option. Repeat for Companies B and C, and explain why this makes sense.
Answer: When the number of miles traveled is less than 100, Company A will be least expensive. If the number of miles traveled is between 100 and 350, then Company B will be least expensive. If the number of miles traveled is greater than 350, then Company C will be the least expensive option.
6. In 2003, the number N of cases of SARS (Severe Acute Respiratory Syndrome) reported in Hong Kong was initially approximated by N = 78.9 + 30.1t, where t is the number of days since March 17. Give the units on, and interpret the meanings of, the constants 78.9 and 30.1.
Answer: The number 78.9 has units of cases and represents the number
Answer: q = 3 + 2t. See the graph below.
(b) A quantity q begins at 8 and decreases by 3 per day.
Answer: q = 8 − 3t. See the graph below.
(c) It has a small negative intercept and a large positive slope.
Answer:
(d) It has a large positive intercept and a small negative slope.
Answer:
