Concept explainers
Shannon Diversity IndexOne way to measure species diversity is to use the Shannon diversity index
where
to show that the maximum value of
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Chapter 13 Solutions
Multivariable Calculus
- Mass-Luminosity Relation Roughly 90 of all stars are main-sequence stars. Exceptions include supergiants, giants, and dwarfs. For main-sequence stars including the sun, there is an important relationship called the mass-luminosity relation between the relative luminosity L and the mass M in terms of solar masses. Relative masses and luminosities of several main-sequence stars are reported in the accompanying table. Star Solar mass M Luminosity L Spica 7.3 1050 Vega 3.1 55 Altair 1 1.1 The Sun 1 1 61 Cygni A 0.17 0.002 a Find a power model for the data in this table. Round the power and the coefficient to one decimal place. The function you find is known to astronomers as the mass-luminosity relation. b Kruger 60 is a main sequence star that is about 0.11 solar mass. Use functional notation to express the relative luminosity of Kruger 60, and then calculate that value. c Wolf 359 has a relative luminosity of about 0.0001. How massive is Wolf 359? d If one star is 3 times as massive as another, how do their luminosities compare?arrow_forwardDecay 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_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