Concept explainers
Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If L1(t) and L2(t) are the sizes of two organs in an organism of age t, then L1 and L2 satisfy an allometric law if their specific growth rates are proportional:
where k is a constant.
(a) Use the allometric law to write a differential equation relating L1 and L2 and solve it to express L1 as a function of L2.
(b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating B (cell biomass) and V (cell volume) was found to be k = 0.0794. Write B as a function of V.
Trending nowThis is a popular solution!
Chapter 9 Solutions
Calculus: Early Transcendentals
- 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.arrow_forwardHubbles Constant Astronomers believe that the universe is expanding and that stellar objects are moving away from us at a radial velocity V proportional to the distance D from Earth to the object. a. Write V as a function of D using H as the constant of proportionality. b. The equation in part a was first discovered by Edwin Hubble in 1929 and is known as Hubbles law. The constant of proportionality H is known as Hubbles constant. The currently accepted value of Hubbles constant is 70 kilometers per second per megaparsec. One megaparsec is about 3.0861019 kilometers. With these units for H, the distance D is measured in megaparsecs, and the velocity V is measured in kilometers per second. The galaxy G2237+305 is about 122.7 megaparsecs from Earth. How fast is G2237+305 receding from Earth? c. One important feature of Hubbles constant is that scientists use it to estimate the age of the universe. The approximate relation is y=1012H Where y is time in years. Hubbles constant is extremely difficult to measure, and Edwin Hubbles best estimate in 1929 was about 530 kilometers per second per megaparsec. What is the approximate age of the universe when this value of H is used? d. The calculation in part c would give scientists some concern, since Earth is thought to be about 4.6 billion years old. What estimate of the age of the universe does the more modern value of 70 kilometers per second per megaparsec give?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage