Concept explainers
Reminder Round all the answers to two decimal places unless otherwise indicated.
Sales Growth A class of models for population growth rates in marine fisheries assumes that the harvest from fishing is proportional to the population size One such model uses a quadratic function:
Here
a. Make a graph of
b. Calculate
c. At what population size is the growth rate the largest?
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Alligators Refer to Exercise 93. If an alligator weighs 125 pounds, what is its approximate length? Round to the nearest inch. 93. Alligators The length in inches and weight in pounds of 25 alligators is shown in the table. Find the quadratic function that best fits the data. Round a,b, and c to six decimal places. Use the regression function to estimate the weight of an alligator that is 130 inches long. Round the weight to the nearest pound. Length Weight Length Weight Length Weight 94 130 72 38 90 106 74 51 128 366 89 84 147 640 85 84 68 39 58 28 82 80 76 42 86 80 86 83 114 197 94 110 88 70 90 102 63 33 72 61 78 57 86 90 74 54 69 36 61 44arrow_forwardFlood Control A river is 8 feet above its flood stage. The water is receding at a rate of 3 inches per hour. Write a mathematical model that shows the number of feet above flood stage after t hours. Assuming the water continually recedes at this rate, when will the river be 1 foot above its flood stage?arrow_forwardTEST YOUR UNDERSTADING FOR EXAMPLE 3.5 A donation to the university is required for the privilege of purchasing premium season football passes. The cost is a linear function of the number of season passes. It costs 8000 to get season passes for myself and my spouse. To get passes for my entire family of 5, it costs 17,000. Use a formula to express the total cost C, in dollars, of n season passes. EXAMPLE 3.5 CHANGING CELSIUS TO FAHRENHEIT Temperature F=F(C) in Fahrenheit is a linear function of the temperature C in Celsius. A lab assistant placed a Fahrenheit thermometer beside a Celsius thermometer and observed the following: When the Celsius thermometer reads 30 degree (C=30), the Fahrenheit thermometer reads 86 degrees (F=86). When the Celsius thermometer reads 40 degrees, the Fahrenheit thermometer reads 104 degrees. Part 1 Use a formula to express F as a linear function of C. Part 2 At sea level, water boils at 212 degrees Fahrenheit. What temperature in degrees Celsius makes water boil? Part 3 Explain in practical terms what the slope means in this setting.arrow_forward
- Working on a Commission A certain man works in sales and earns a base salary of 1000 per month plus 5 of his total sales for the month. a. Explain why his total monthly income I is a linear function of total sales S, both measured in dollars. b. How much does he earn if he sells 1600 in merchandise in a month? c. Write a formula that gives total monthly income as a linear function of sales in a month. d. What should his monthly total sales be if he wishes to earn 1350 this month?arrow_forwardMarine Fishery One class of models for population growth rates in marine fisheries assumes that the harvest from fishing is proportional to the population size. For one such model, we have G=0.3n(1n2)0.1n Here G is the growth rate of the population, in millions of tons of fish per year, and n is the population size, in millions of tons of fish. a.Make a graph of G versus n. include values of n up to 1.5 million tons. b.Use functional notation to express the growth rate if the population size is 0.24 million tons, and then calculate that value. c. Calculate G1.42 and explain in practical terms what your answer means. d.At what population size is the growth rate the largest?arrow_forwardTraffic Accidents The following table shows the cost C of traffic accidents. in cents per vehicle-mile, as a function of vehicular speed s, in miles per hour, for commercial vehicles driving at night on urban streets. Speed s 20 25 30 35 40 45 50 Cost C 1.3 0.4 0.1 0.3 0.9 2.2 5.8 The rate of vehicular involvement in traffic accidents per vehicle-mile can be modeled as a quadratic function of vehicular speed s, and the cost per vehicular involvement is roughly a linear function of s, so we expect that C the product of these two functions can be modeled as a cubic function of s. a. Use regression to find a cubic model for the data. Keep two decimal places for the regression parameters written in scientific notation. b. Calculate C(42) and explain what your answer means in practical terms. c. At what speed is the cost of traffic accidents for commercial vehicles driving at night on urban streets at a minimum? Consider speeds between 20 and 50 miles per hour.arrow_forward
- Size of High Schools The farm population has declined dramatically in the years since World War II, and with that decline, rural school districts have been faced with consolidating in order to be economically efficient. One researcher studied data from the early 1960s on expenditures for high schools ranging from 150 to 2400 in enrollment. He considered the cost per pupil as a function of the number of pupils enrolled in the high school, and he found the approximate formula C=7430.402n+0.00012n2 where n is the number of pupils enrolled and C is the cost, in dollars, per pupil. a. Make a graph of C versus n. b. What enrollment size gives a minimum per-pupil cost? c. If a high school had an enrollment of 1200, how much in per-pupil cost would be saved by increasing enrollment to the optimal size found in part b?arrow_forwardTotal Cost The background for this exercise can be found in Exercises 13 and 14 in Section 3.2. The following table gives the total cost C, in dollars, for a widget manufacturer as a function of the number N of widgets produced during a month. Number N Total cost C 200 7900 250 9650 300 11, 400 350 13, 150 a. What are the fixed costs and variable cost for this manufacturer? b. The manufacturer wants to reduce the fixed costs so that the total cost at a monthly production level of 350 will be 12, 975. What will the new fixed costs be? c. Instead of reducing the fixed costs as in part b, the manufacturer wants to reduce the variable cost so that the total cost at a monthly production level of 350 will be 12, 975. What will the new variable cost be?arrow_forward
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