Concept explainers
Automobile Aerodynamics The horsepower H required to overcome wind drag on a particular automobile is given by
(a) Use a graphing utility to graph the
(b) Rewrite the horsepower function so that x represents the speed in kilometers per hour.
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- Reaction Rates In a chemical reaction, the reaction rate R is a function of the concentraton of the product of the reaction. For a certain second-order reaction between two substances, we have the formula R=0.01x2x+22. Here x is measured in moles per cubic meter and R is measured in moles per cubic meter per second. a. Make a graph of R versus x. Include concentrations up to 100 moles per cubic meter. b. Use functional notation to express the reaction rate when the concentration is 15 moles per cubic meter, and then calculate hat value. c. The reaction is said to be in equilibrium when the reaction rate is 0. At what two concentratoins is the reaction in equilibrium?arrow_forwardBungee Jumping Luisa goes bungee jumping from a 500-ft-high bridge. The graph shows Luisas height ht in ft after t seconds. Describe in words what the graph indicates about Luisas bungee jump. Suppose Luisa goes bungee jumping from a 400-ft-high bridge. Sketch a new graph that shows Luisas height Ht after t seconds. What transformation must be performed on the function h to obtain the function H? Express the function H in terms of h.arrow_forwardMagazine Circulation: The circulation C of a certain magazine as a function of time t is given by the formula C=5.20.1+0.3t Here C is measured in thousands, and t is measured in years since the beginning of 2006, when the magazine was started. a. Make a graph of C versus t covering the first 6 years of the magazines existence. b. Express using functional notation the circulation of the magazine 18 months after it was started, and then find that value. c. Over what time interval is the graph of C concave up? Explain your answer in practical terms. d. At what time was the circulation increasing the fastest?. e. Determine the limiting value for C. Explain your answer in practical terms.arrow_forward
- Population Growth: The growth G of a population over a week is a function of the population size n at the beginning of the week. If both n and G are measured in thousands of animals, the formula is G=0.25n2+5n, a. Make a graph of G versus n. Include values of n up to 25 thousand animals. b. Use functional notation to express the growth over a week if the population at the beginning is 4 thousand animals, and then calculate that value. c. Calculate G(22) and explain in practical terms what your answer means. d. For what values of n is the function G increasing? Determine whether the graph is concave up or concave down for these values, and explain in practical terms what this means.arrow_forwardResale Value: The resale value V, in dollars, of a certain car is a function of the number of year t since the year 2012. In the year 2012, the resale ale is 18,000 and each year thereafter the resale value decreases by 1700. a. What is the resale value in the year 2013? b. Find a formula for V as a function of t. c. Make a graph of V versus t covering the first 4 years since the year 2012. d. Use functional notation to express the resale value in the year 2015, and then calculate that value.arrow_forwardPlanet Growth The amount of growth of plants in an ungrazed pasture is a function of the amount of plant biomass already present and the amount of rainfall. For a pasture in the arid zone of Australia the formula Y=55.120.01535N0.00056N2+3.946R gives an approximation of the growth. Here R is the amount of rainfall, in millimeters, over a 3 month period; N is the plant biomass, in kilograms per hectare, at the beginning of that period; and Y is the growth, in kilograms per hectare, of the biomass over that period. For comparison, 100 millimeters is about 3.9 inches, and 100 kilograms per hectare is about 89 pounds per acre. For this exercise, assume that the amount of plant biomass initially present is 400 kilograms per hectare, so N=400. a. Find a formula for the growth Y as a function of the amount R of rainfall. b. Make a graph of Y versus r. Include values of R from 40 to 160 millimeters. c. What happens to Y as R increases? Explain your answer in practical terms. d. How much growth will there be over a 3 month period if initially there are 400 kilograms per hectare of plant biomass and the amount of rainfall is 100 millimeters?arrow_forward
- Fluid Flow The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine whether the input pipe and each drainpipe are open or closed in specific subintervals of the 1 hour of time shown in the graph. (There are many correct answers.)arrow_forwardPopulation Growth The projected population of the United States for the years 2025 through 2055 can be modeled by P=307.58e0.0052t, where P is the population (in millions) and t is the time (in years), with t=25 corresponding to 2025. (a) Use a graphing utility to graph the function for the years 2025 through 2055. (b) Use the table feature of the graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, during what year will the population of the United States exceed 430 million?arrow_forwardSpawner-Recruit Model In fish management it is important to know the relationship between the abundance of the spawners also called the parent stock and the abundance of the recruitsthat is, those hatchlings surviving to maturity. According to the Ricker model, the number of recruits R as a function of the number of spawners P has the form R=APeBp for some positive constants A and B. This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits. In a study of the sockeye salmon, it was determined that A=4 and B=0.7. Here we measure P and R in thousands of salmon. a. Make a graph of R versus P for the sockeye salmon. Assume there are at most 3000 spawners. b. Find the maximum number of salmon recruits possible. c. If the number of recruits R is greater than the number of spawners P, then the difference R-P of the recruits can be removed by fishing, and next season there will once again be P spawners surviving to renew the cycle. What value of P gives the maximum value of R-P, the number of fish available for removal by fishing?arrow_forward
- a. Use a formula to express the altitude of a helicopter as a function of time. Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the altitude of the helicopter 90 seconds after takeoff, and then calculate that value. c. Make a graph of altitude versus time covering the first 3 minutes of the flight. Explain how the description of the function is reflected in the shape of the graph.arrow_forwardBird Population The graph shows the population of a rare species of bird, where t represents years since 2009 and n(t) is measured in thousands. a Find a function that models the bird population at time t in the form n(t)=n0ert. b What is the bird population expected to be in the year 2020?arrow_forwardTax Owed The income tax T owed in a certain state is a function of the taxable income I, both measured in dollars. The formula is T=0.11I500 a. Express using functional notation the tax owed on a taxable income of 13, 000, and then calculate that value. b. If your taxable income increases from 13,000 to 14,000, by how much does your tax increase? c. If your taxable income increases from 14,000 to 15,000, by how much does your tax increase?arrow_forward
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