Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Giants Ants and Spiders Many science fiction movies feature animals such as ants, spiders, or apes growing to monstrous sizes and threatening defenseless Earthlings. (Of course, they are in the end defeated by the hero and heroine.) biologists use power function as a rough guide to relate body weight and cross-sectional area of limbs to length or height. Generally, weight is thought to be proportional to the cube of length, whereas the cross-sectional area of limbs is proportional to the square of length. Suppose an ant, having been exposed “radiation” is enlarged to
a. By how much will its weight be increased?
b. By how much will the cross-sectional area of its legs be increased?
c. Pressure on a limb is weight divided by cross-sectional area. By how much has the pressure on a leg of the giant ant increased? What do you think is likely to happen to this unfortunate ant? (Note: The factor by which pressure increases is given by
.
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- ReminderRound all answers to two decimal places unless otherwise indicated. DensityThe total weight of a rock depends on its size and is proportional to its density. In this context, density is the weight per cubic inch. Let w denote the weight of the rock in pounds, s the size of the rock in cubic inches, and d the density of the rock in pounds per cubic inch. a. What is the total weight of a 3-cubic-inch rock that weighs 2 pounds per cubic inch? b. Write an equation that shows the proportionality relation. What is the constant of proportionality? c. Use the equation you found in part b to find the total weight of a 14-cubic-inch rock with density 0.3 pound per cubic inch.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Bores Under certain conditions, tsunami waves encountering land will develop into bores. A bore is a surge of water much like what would be expected if a dam failed suddenly and emptied a reservoir in to a river bed. In the case of a bore travelling from the ocean into a dry river bed, one study shows that the velocity V of the tip of the bore is proportional to the square root of its height h. Expressed in a formula, this is V=kh0.5, where k is a constant. a.A bore travels up a dry river bed. How does the velocity of the tip compare with its initial velocity when its height is reduced to half of its initial height? b.How does the height of the bore compare with its initial height when the velocity of the tip is reduced to half of its initial velocity? c.If the tip of one bore surging up a dry river bed is three times the height of another, how do their velocities compare?arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Hair Growth When you are 18 years old you have a hair that is 14 centimeters long, and your hair grows about 12 centimeters each year. Let H(t) be the length, in centimeters, of that hair t years after age 18. a. Find a formula that gives H as a linear function of t. b. How long will it take for the hair to reach a length of 90 centimeters?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Kleibers law states that for the vast majority of animals, the metabolic rate M is a power function of the weight W, and the power is k=34. a.The Eastern gray squirrel weighs about 1 pound. How does the squirrels metabolic rate compare with that of a 200 -pound man? b.How does a 200- pound mans metabolic rate compare with that of a 130- pound woman? c.Based on your answer to part b, would overeating the same amount for each be more likely to lead to weight gain for the man or for the woman?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Getting Celsius Fro Fahrenheit Water freezes at 0 degrees Celsius, which is the same as 32 degrees Fahrenheit. Also, water boils at 100 degrees Celsius, which is the same as 212 degrees Fahrenheit. a. Use the freezing and boiling points of water to find a formula expressing Celsius temperature C as a linear function of the Fahrenheit temperature F b. What is the slope of the function you found in part a? Explain its meaning in practical terms. c. In Example 3.5, we showed that F=1.8C+32. Solve this equation for C and compare the answer with that obtained in part a.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_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Estimating Wave Height Sailors use the following function to estimate wave height h, in feet, from wind speed w, in miles per hour h=0.02w2 a. Make a graph of wave height versus wind speed. Include wind speeds of up to 25 miles per hour. b. A small boat can sail safely provided wave heights are no more than 4 feet. What range of wind speed will give safe sailing for this boat?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_forwardReminder 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_forward
- Reminder Round all answer to two decimal places unless otherwise indicated. Lean Body Weight in Males Your lean body weight L is the amount you would weigh if all the fat in your body were to disappear. One text gives the following estimate of lean body weight L in pounds for young adult males: L=98.42+1.08W4.14A, where W is total weight in pounds and A is abdominal circumference in inches. 7 a. Consider a group of young adult males who have the same abdominal circumference. If their weight increases but their abdominal circumference remains the same, how does their lean body weight change? b. Consider a group of young adult males who have the same weight. If their abdominal circumference decreases but their weight stays the same, how does their lean body weight change? c. Suppose a young adult male has a lean body weight of 144 pounds. Over a period of time, he gains 15 pounds in total weight, and his abdominal circumference increases by 2 inches. What is his lean body weight now?arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. Protein Content of Wheat GrainProtein content of wheat grain is affected by soil moisture and the amount of available nitrogen among other things. Figure 1.45 shows" the percent of protein content of wheat grain versus pounds of nitrogen per acre applied in three separate situations. In each case, soil moisture refers to moisture at the soil depth of 2 inches to 12 inches. Situation 1: Irrigation was used when soil moisture dropped to 49. Situation 2: Irrigation was used when soil moisture dropped to 34. Situation 3: Irrigation was used when soil moisture dropped to 1. a. If irrigation begins when soil moisture reaches 49, what application of nitrogen will result in the lowest percentage of protein in wheat grain? b. If irrigation begins when soil moisture reaches 34, what application of nitrogen will result in the same protein content of wheat grain as beginning irrigation when soil moisture reaches 1? c. If you irrigate when soil moisture reaches 34, how much nitrogen should you apply to achieve a 13 protein content in wheat grain? d. Does Figure 1.45 indicate that, for nitrogen levels at 45 pounds per acre or higher, increased protein content in wheat grain is associated with higher or lower soil moisture? FIGURE 1.45 Protein content versus availability of nitrogenarrow_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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning