Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Unit Conversion with Exponential Growth The exponential function
a. What is the yearly growth factor? Find a formula that gives the population
b. What is the century growth factor? Find a formula that gives the U.S. population
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Chapter 4 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Half-Life of Radioactive Iodine A student is trying to determine the half-life of radioactive iodine-131. He measures the amount of iodine-131 in a sample solution every 8 hours. His data are shown in the table below. a Make a scatter plot of the data. b Use a calculator to find an exponential model. c Use your model to find the half-life of iodine-131. Time h Amount of 131I(g) 0 8 16 24 32 40 48 4.80 4.66 4.51 4.39 4.29 4.14 4.04arrow_forwardRound each percentage increase or decrease that you calculate to the nearest tenth of a percent. Percentage Decay Rate from Decay FactorAn exponential function has a decay factor of 0.70. What is the percentage decay rate?arrow_forwardRound each percentage increase or decrease that you calculate to the nearest tenth of a percent. Percentage Growth Rate from Growth FactorAn exponential function has a growth factor of 1.06. What is the percentage growth rate?arrow_forward
- A Skydiver If a skydiver jumps from an airplane, his velocity v, in feet per second, starts at 0 and increases towards terminal velocity, an average-sized man has a terminal T of about 176 feet per second. The difference D=Tv is an exponential function of time, a. What is the initial value of D? b. Two second into fall, the velocity is 54.75 feet per seconds. Find an exponential formula for D. c. Find a formula for v. d. Express the velocity 4 seconds into the fall using functional notation, and then calculate that value.arrow_forwardChanging UnitsA certain quantity has a yearly decay factor of 0.99.What is its century decay factor?arrow_forwardYou invest $6000 at an annual rate of 4. Find the balance after 7 years for each type of compounding. a.Quarterlyb.Monthlyc.Continuousarrow_forward
- The function f(x)=5x is an exponential function with base______; f(2) =_______, f(0) =______, f(2) =_______, and f(6) =_______.arrow_forwardTEST YOUR UNDERSTANDING| FOR EXAMPLE 4.2 In a small country, the population is initially 1.4million. To find next years population from this years population, we multiply by 1.02. Find an exponential formula that gives the population N, in millions, after tyears. When does the population reach 6millions?arrow_forwardGrowth Rate Versus Weight Ecologists have studied how a populations intrinsic exponential growth rate r is related to the body weight W for herbivorous mammals. In table 5.2, W is the adult weight measured in pounds, and r is growth rate per year. Animal Weight W r Short-tailed vole 0.07 4.56 Norway rat 0.7 3.91 Rue deer 55 0.23 White-tailed deer 165 0.55 American elk 595 0.27 African elephant 8160 0.06 Find a formula that models r as a power function of W, and draw a graph of this function.arrow_forward
- TEST YOUR UNDERSTANDING | FOR EXAMPLE 5.10 When the cake in the example reaches a temperature of 300 degrees, it is baked. The cake is taken from the oven and placed on a counter to cool. The temperature of the room is 75degrees. The difference between the cakes temperature and room temperature is an exponential function of time, in minutes, with decay factor 0.85. What is the initial value of this exponential function? What is the limiting value of the cakes temperature? Find a formula that gives the cakes temperature after t minutes. EXAMPLE 5.10BAKING A CAKE A pan of cake batter is initially at a room temperature of 75degrees. The pan is placed in a 350degree oven to bake. Let C=C(t) denote the temperature of the cake batter tminutes after it is placed in the oven. In this example, all temperatures are measured in degrees Fahrenheit. Part 1 The temperature of the cake batter is given by C=LimitingvalueTemperature difference. What is the limiting value of C. Part 2 Let D denote the difference between the oven temperature and that of the batter. Then Newtons law of cooling tells us that D is an exponential function. What is the initial value of D. Part 3 After 10 minutes, we find that the cake batter has heated to 165 degrees. Find a formula for D. Part 4 Find a model for the temperature of the cake batter t minutes after it is placed in the oven.arrow_forwardUse a calculator to help solve each problem. Tritium decay Tritium, a radioactive isotope of hydrogen, has a half-life of 12.4 years. Of an initial sample of 50 grams, how much remain after 100 years?arrow_forward
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