Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Catering a Dinner You are having a dinner catered. You pay a rental fee of
a. Suppose you just want to break even.
i. How much should you charge per ticket if you expect
ii Use a formula to express the amount you should charge per ticket as a function of the number of people attending. Be sure to explain the meaning of the letters you choose and the units.
iii. You expect
b. Suppose now that you want to make a profit of
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
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- Reminder Round all answers to two decimal places unless otherwise indicated. A wedding Reception You rent a wedding venue for a cost of 3200. The cost includes a catered lunch for 50 guests. For each addition guest, though, the catered lunch costs 31. a. What is the cost of the venue and lunch if you invite 100 guests. b. Find a formula showing the cost of the venue and the lunch as a function of n, the number of guests. Assume that n is at least 50. c. The amount you have budgeted for the venue and catered lunch is 5500. How many guests can you invite.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. World Copper Production World production of copper, in millions of tons per year, from 1900 to 2000 is given by C=0.51.033t, where t is the time in years since 1900. a.What production level does this model give for the year 2000? b.If this model were extended to 2025, how could you use your knowledge of copper production in 2024 to estimate copper production in 2025?arrow_forwardReminder: Round all answer to two decimal places unless otherwise indicated. a. How much does it cost to prepare and mail a 3-page letter if your secretary spends 2 hours on typing and corrections? b. Use a formula to express the cost of preparing and mailing a letter as a function of the number of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. c. Use the function you made in part b to find the cost of preparing and mailing a 2-page letter that it takes your secretary 25 minutes to type. Note: 25 minutes is 25/60 hour.arrow_forward
- Reminder Round all answer to two decimal places unless otherwise indicated. Real Estate Sales A real estate agency has fixed monthly costs associated with rent, staff salaries, utilities, and supplies. It earns its money by taking a percentage commission on total real estate sales. During the month of July, the agency had total sales of 832,000 and showed a net income after paying fixed costs of 15,704. In August, total sales were 326,000 with a net income of only 532. a. Use a formula to express net income as a linear function of total sales. Be sure to identify what the letters that you use mean. b. Plot the graph of net income and identify the slope and vertical intercept. c. What are the real estate agencys fixed monthly costs? d. What percentage commission does the agency take on the sale of a home? e. Find the horizontal intercept and explain what this number means to the real estate agency.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Nail Growth The rate of fingernail growth depends on many factors, but in adults, nails grow at an average rate of 3 millimeters per month. If a nail is initially 12 millimeters long, find a formula that gives the length L, in millimeters, of the nail if left unclipped after t months.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Profit The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. In this exercise, we measure all monetary values in dollars. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering such things as plant maintenance and insurance as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. The total revenue R for a manufacturer during a given time period is also a function of the number N of items produced during that period. To determine a formula for the total monthly revenue, we need to know the selling price p per unit of the item, which in general is also a function of N. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. Suppose that a manufacturer of widgets has a fixed costs of 2000 per month and that the variable cost is 30 per widget. Further, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. Number N Price p 200 41.00 250 40.50 300 40.00 350 39.50 a.Use a formula to express the total monthly cost C of this manufacturer as a function of N. b.Use the table to find a linear model of p as a function of N. c.Use your answer to part b to find a formula expressing the total monthly revenue R as function of N. d.Use your answers to part a and part c to find a formula expressing the monthly profit P as a function of N. What type of function is the profit: linear or quadratic? e.Find the two monthly production levels at which the manufacturer just breaks even that is, where the profit is zero.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Insect ControlDDT dichlorodiphenyltrichloroethane was used extensively from 1940 to 1970 as an insecticide. It still sees limited use for control of disease. But DDT was found to be harmful to plants and animals, including humans, and its effects were found to be lasting. The amount of time that DDT remains in the environment depends on many factors, but the following table shows what can be expected of 100 kilograms of DDT that has seeped into the soil. t=time,inyearssinceapplication D=DDTremaining,inkilograms 0 100.00 1 95.00 2 90.25 3 85.74 a. Show that the data are exponential. b. Make a model of D as an exponential function of t. c. What is the half-life of DDT in the soil? That is, how long will it be before only 50 kilograms of DDT remain?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Paying Off a Credit Card You owe 8000 on a credit card. Because of the high balance, you stop charging and begin paying off the card. You can afford to make only the minimum monthly payment, which is 5 of the balance. The card carries an APR of 24. Under these conditions, the remaining balance alter t monthly payments is given by B=8000(1.020.95)tdollars, You feel comfortable resuming charging with the card when the balance is between 1000 and 2000. a. Plot the graph of B along with the target values of 1000 and 2000. Use a horizontal span of 0 to 100. b. For what values oft is the function B in the desired range? c. After how many monthly payments can you resume charging? Your answer should be a whole number.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Renting Motel Rooms You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for 85.00, two for 83.00 each, and in general the group rate per room is found by taking 2 off the base of 85 for each extra room rented. a.How much money do you charge per room if a group rents 3 rooms? What is the total amount of money you take in? b.Use a formula to give the rate you charge for each room if you rent n rooms to an organization. c.Find a formula for a function R=R(n) that gives the total revenue from renting n rooms to a convention host. d.Use functional notation to show the total revenue from renting a block of 9 rooms to a group. Calculate that value.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Walking and Running You live east of campus, and you are walking from campus toward your home at a constant speed. When you get there, you rest for 5minutes and then run back west at a rapid speed. After a few minutes, you reach your destination, and then you rest for 10minutes. Measure your location as your distance west of your home, and make graphs of your location and velocity.arrow_forwardReminder: Round all answer to two decimal places unless otherwise indicated. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To deter mine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost C of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. InflationDuring a period of high inflation, a political leader was up for re-election. Inflation had been increasing during his administration, but he announced that the rate of increase of inflation was decreasing. Draw a graph of inflation versus time that illustrates this situation. Would this announcement convince you that economic conditions were improving?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning