EXAMPLE 4 Making a Linear Model from a Rate of Change Water is being pumped into a swimming pool at the rate of 5 gal per min. Initially, the pool contains 200 gal of water. (a) Find a linear function V that models the volume of water in the pool at any time t. (b) If the pool has a capacity of 600 gal, how long does it take to completely fill the pool? There are 200 gallons of water in the pool at time t = 0.

EXAMPLE 4 Making a Linear Model from a Rate of Change Water is being pumped into a swimming pool at the rate of 5 gal per min. Initially, the pool contains 200 gal of water. (a) Find a linear function V that models the volume of water in the pool at any time t. (b) If the pool has a capacity of 600 gal, how long does it take to completely fill the pool? There are 200 gallons of water in the pool at time t = 0.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section2.5: Linear Functions And Models

Problem 42E: Filling a Pond A large koi pond is filled from a garden hose at the rate of 10 gal/min. Initially,...

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Filling a PondA large koi pond is filled from a garden hoseat the rate of 10 gal/min.Initially,the pond contains 300 galof water. (a) Find a linear function V that models the volume of waterin the pond at any time t . (b) If the pond has a capacity of 1300 gal, how long does it take to completely fill the pond?
The number of people afflicted with the common cold in the winter months dropped steadily by 50 each yearsince 2004 until 2010. In 2004, 875 people were inflicted. Find the linear function that models the number of people afflicted with the common cold C as a function of theyear, t. When will no one be afflicted?
Flood 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?
Grazing Kangaroos The amount of vegetation eaten in a day by a grazing animal V of food available measured as biomass, in units such as pounds per acre. This relationship is called the functional response. If there is little vegetation available, the daily intake will be small, since the animal will have difficulty finding and eating the food. As the amount of food biomass increases, so does the daily intake. Clearly, though, there is a limit to the amount the animal will eat, regardless of the amount of food available. This maximum amount eaten is the satiation level. a.For the western grey kangaroo of Australia, the functional response is G=2.54.8e0.004V, where G=G(V) is the daily intake measured in pounds and V is the vegetation biomass measured in pounds per acre. i. Draw a graph of G against V. Include vegetation biomass levels up to 2000 pounds per acre. ii. Is the graph you found in part i concave up or concave down? Explain in practical terms what your answer means about how this kangaroo feeds. iii. There is a minimal vegetation biomass level below which the western grey kangaroo will eat nothing. Another way of expressing this is to say that the animal cannot reduce the food biomass below this level. Find this minimal level. iv. Find the satiation level for the western grey kangaroo. b. For the red kangaroo of Australia, the functional response is R=1.91.9e0.033V, Where R is the daily intake measured in pounds and V is the vegetation biomass measured in pounds per acre. i. Add the graph of R against V to the graph of G you drew in part a. ii. A simple measure of the grazing efficiency of an animal involves the minimal vegetation biomass level described above: The lower the minimal level for an animal, the more efficient it is at grazing. Which is more efficient at grazing, the western grey kangaroo or the red kangaroo?
Filling a Pond A large koi pond is filled from a garden hose at the rate of 10 gal/min. Initially, the pond contains 300 gal of water. a Find a linear function V that models the volume of water in the pond at any time t. b If the pond has a capacity of 1300 gal, how long does it take to completely fill the pond?
Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.