- During the first and second quarters of a year, a business had sales of $42,000 and $58,000, respectively. If the growth of sales follows a linear pattern for the next four years, what will sales be in the fourth quarter? In the 9th quarter? Use an algebraic method of solution.
- Ajax Manufacturing bought a machine for $48,000. It is expected to last 15 years and, at the end of that time, have a salvage value of $7,000. Set up a linear depreciation model for this machine and find the worth at the end of 10 years.
- For a certain type of letter sent by Federal Express, the charge is $8.50 for the first 8 ounces and $0.90 for each additional ounce (up to 16 ounces.) How much will it cost to send a 12-ounce letter?
- Write a mathematical model for the population of this city over the given period of time and use that to predict the population in 2010.
- When cigarettes are burned, one by-product in the smoke is carbon monoxide. Data is collected to determine whether the carbon monoxide emission can be predicted by the nicotine level of the cigarette. It is determined that the relationship is approximately linear when we predict carbon monoxide, C, from the nicotine level, N. Both variables are measured in milligrams. The formula for the model is .
- Interpret the slope.
- Interpret the intercept.
- Reinforced concrete buildings have steel frames. One of the main factors affecting the durability of these buildings is carbonation of the concrete (caused by a chemical reaction that changes the pH of the concrete) which then corrodes the steel reinforcing the building. Data is collected on specimens of the core taken from such buildings, where the depth, d, of the carbonation, in mm, and the strength, s, of the concrete, in mega-Pascals (MPa,) are measured. It is found that the model is
- Interpret the slope.
- Interpret the intercept.
In addition to learning to work applied problems, one point of this lesson is to increase your ease and flexibility in reading and working problems. For that reason, some of the problems are stated somewhat differently than problems in the examples. Ask questions as needed.
Notice that many of the following problems have parts labeled with letters. In your solution, you are required to label each of the parts of your solution with the appropriate letter that shows which question you are answering.
Sometimes students prefer to work problems using exactly the same steps as the examples, numbering the steps in the same way. If you would prefer to work the problems in that way, then first do that, numbering the steps just as in the examples. AFTER you have completed that, then re-read the problem and go back and put additional labels onto your solution, to indicate where each part of the stated problem is solved.
- A college had linear growth in enrollment over the period from 1993 – 2003. In 1997 they had 6754 students enrolled and in 2000 they had 8117 students enrolled. If the same pattern in growth continues, how many students do you expect they will have in 2010?
- Let t = number of years since 1993. Let E = enrollment. Make a table by hand or in a spreadsheet that shows the relationship of E to t over the given period of time.
- Graph the relationship.
- Write a linear model for the relationship of E to t.
- Interpret the slope and y-intercept in the formula, using the units in the problem.
- Use the formula to predict the enrollment in 2010.
- Use your graph to predict the enrollment in 2010.
- New question—not stated in the original problem: Approximately when do we expect the enrollment to be 11,380 students? Use either your graph or formula to do this. Explain which you used and how you did it.
- A person deposits a certain amount of money in an account that pays simple interest.Thus the amount of money in the account at any time is a linear function of time. After 2 months, the amount in the account is $759.After 3 months, the amount in the account is $763.50.
- Do all the steps necessary to find a linear model to relate the amount in the account, y, to the number of months, x. Be sure to interpret the slope and intercept.
- Use your formula for the linear model to find the amount that will be in the account after 36 months.
- Use a spreadsheet to make a table of the amount in the account after each month up to 36 and check the answer you obtained using algebra.
- One can measure temperature in degrees Celsius or degrees Fahrenheit. The two measurements are linearly related. The temperature at which water freezes is 0 degrees Celsius and 32 degrees Fahrenheit. The temperature at which water boils is 100 degrees Celsius and 212 degrees Fahrenheit. We want to predict the temperature Celsius.
- Let C = degrees Celsius and F = degrees Fahrenheit. Do all the steps necessary to find a linear model to describe this relationship.
- Interpret the slope and y-intercept of equation, using the units in the problem.
- When the temperature is 72 degrees Fahrenheit, what is the Celsius temperature?
- Using your formula, make a table that shows the temperature Fahrenheit from –20 degrees to 120 degrees and the predicted Celsius temperature for each of these. If you can use a spreadsheet for this, produce a table that goes in increments of 1 degree. If you are doing it by hand, produce a table that goes in increments of 10 degrees.
- Graph this relationship.
- In 1991, the number of outlet shopping centers in the US was 142. By 1993, the number had increased to 249.
(Hint: Let t = years since 1991 and then solve the problem using t.)
- If the number of outlet shopping centers continued to increase in a linear pattern, what would have been the number in 1994? Find a linear model and use it to make the prediction.
- In fact, the actual number of outlet shopping centers in 1994 was 300. Do you think that the increase in the number of outlet shopping centers was approximately linear?
- Make a graph of the data given in the initial set-up of the problem. Draw the line. Extend the line to 1994. What does your line predict the number of outlet shopping centers will be in 1994? Does that agree with the prediction your formula gave in part a?
For problems 11–14, write an algebraic linear model, interpret the slope and y-intercept, use the linear model to answer the question, and make a graph to check your work.
- A cellular phone company has equipment that can service 80 thousand customers. In 2000 they had 57 thousand customers and, over the last few years, they have been adding about 3,000 customers per year. How many customers will they have in 2006? If this rate of increase continues, when will they need additional equipment?
- The manager of a supermarket finds that she can sell 1130 gallons of milk per week at $3.99 per gallon and 1470 gallons of milk per week at $3.79. Assume that the sales, s, is a linear formula of the price, p. How many gallons would she expect to sell at $3.92 per gallon? (Hint: When you interpret the slope, it may seem strange to you. You might want to re-work the problem using cents instead of dollars for the price. That will make it easier to understand the interpretation of the slope.)
- At 680 Fahrenheit, a certain species of cricket chirps 124 times per minute. At 400 Fahrenheit, the same cricket chirps 86 times per minute. Assume the chirps per minute, C, is linearly related to the temperature Fahrenheit, T. How many times per minute will the cricket chirp at 700 Fahrenheit? If you count the cricket chirps for a minute and find that it is 110 chirps, what is the temperature, to the nearest whole degree?
- A bicycle manufacturer has daily fixed costs of $1500 and each bicycle costs $80 to manufacture. What is the cost of manufacturing 16 bicycles in a day? How many bicycles could be manufactured in one day for $3220?
- Recall the example from Topic B. Formulas on “break-even” analysis. A company sells a catnip cat toy for $3 each and sells all that are produced. The fixed cost of production is $6000 and the variable cost is $1.20 each.
- Write a formula for the cost of producing x toys. [ANSWER:Cost = 6000 + 1.2 x]
- Write another formula for the revenue produced by selling x toys. [ANSWER:Revenue = 3 x]
- Use a spreadsheet to graph both formulas.
- Look at your graph to find the value of x for which the cost is equal to the revenue? (The point on the graph is called the “break-even” point.) [ANSWER: Breakeven is where cost and revenue lines cross.]
- Use algebra to find the x-value of the break-even point and then use one of the formulas to find the y-value of the break-even point. Write your conclusion in a sentence.
- A contractor purchases a piece of equipment for $65,000. The operating cost is $4.63 per hour (electricity, maintenance, etc.) and this year the operator is paid $15.37 per hour. That means that the total operating cost is $(4.63 + 15.37) per hour.
- Find a formula for the cost of the running the machine, C, where the input variable is t = number of hours run. (Be sure to include the purchase cost as the amount it will cost to run it zero hours. Do you see why that makes sense?)
- If the product generated by the machine in per hour is sold for $50, write a formula for the revenue, R, where the input variable is the number of hours run.
- If the machine is run for 1500 hours, what will the cost of running it be?
- If the machine is run for 1500 hours, what will the revenue from running it be?
- Using a spreadsheet, graph both the cost and the revenue formulas on the same axes. (You will need to decide on an appropriate set of t-values to use for your graph. You may need to do some and then extend it to more t-values.)
- For what t-value do the graphs of the formulas intersect? (Find it on the graph. Then use algebra to confirm that you have found it correctly.) Write a sentence to answer the question “How long will they have to run the machine to ‘break even,’ meaning that the revenue will equal the total cost?”
- Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution