Marginal Productivity A manufacturer estimates that pro-duction (in hundreds of units) is a function of the amounts xand y of labor and capital used, as follows.3f(x, y)=(a) Find the number of units produced when 16 units of laborand 81 units of capital are utilized.(b) Find and interpret f,(16, 81) and f,(16, 81).(c) What would be the approximate effect on productionof increasing labor by 1 unit from 16 units of labor with81 units of capital?

The equation which models the production as function of units of labor and capital used is given by,…

Answered: Marginal Productivity A manufacturer…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 63E

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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.
Maximum Sales Growth This is a continuation of Exercise 10. In this exercise, we determine how the sales level that gives the maximum growth rate is related to the limit on sales. Assume, as above, that the constant of proportionality is 0.3, but now suppose that sales grow to a level of 4 thousand dollars in the limit. a. Write an equation that shows the proportionality relation for G. b. On the basis of the equation from part a, make a graph of G as a function of s. c. At what sales level is the growth rate as large as possible? d. Replace the limit of 4 thousand dollars with another number, and find at what sales level the growth rate is as large as possible. What is the relationship between the limit and the sales level that gives the largest growth rate? Does this relationship change if the proportionality constant is changed? e. Use your answers in part d to explain how to determine the limit if we are given sales data showing the sales up to a point where the growth rate begins to decrease.
aThe average rate of change of a function f between x=a and x=b is the slope of the ___________ line between (a,f(a)) and (b,f(b)).
bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.

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