Question

Suppose the output of a certain country is given by $$ f(x, y)=100 x^{3 / 5} y^{2 / 5} $$ billion dollars if $x$ billion dollars are spent on labor and $y$ billion dollars are spent on capital. Find the output if the country spent $\ 32$ billion on labor and $\ 243$ billion on capital.

Name: Problem 47, Chapter 8: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-06-02T05:59:05-08:00
Duration: 1 min 20 s
Channel: Aman Gupta
Description: Suppose the output of a certain country is given by f(x, y)=100 x^3 / 5 y^2 / 5 billion dollars if x billion dollars are spent on labor and y billion dollars are spent on capital. Find the output if the country spent 32 billion on labor and 243 billion on capital.

   Suppose the output of a certain country is given by
$$
f(x, y)=100 x^{3 / 5} y^{2 / 5}
$$
billion dollars if $x$ billion dollars are spent on labor and $y$ billion dollars are spent on
capital. Find the output if the country spent $\ 32$ billion on labor and $\ 243$ billion on capital.

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan 10th Edition

Chapter 8, Problem 47

Instant Answer

Solved by Expert Aman Gupta

06/02/2021

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The function is given by: \[ f(x, y) = 100 x^{3/5} y^{2/5} \] Show more…

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Suppose the output of a certain country is given by $$ f(x, y)=100 x^{3 / 5} y^{2 / 5} $$ billion dollars if $x$ billion dollars are spent on labor and $y$ billion dollars are spent on capital. Find the output if the country spent $\ 32$ billion on labor and $\ 243$ billion on capital.

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Key Concepts

Cobb-Douglas Production Function

The Cobb-Douglas production function is a standard economic model used to represent the relationship between two or more inputs (typically labor and capital) and the produced output. It uses multiplicative inputs raised to constant exponents, which represent the elasticities of output with respect to each input. This type of function helps in analyzing the contribution of each input to total production and is useful in studying the effects of scaling inputs.

Properties of Exponents

Working with functions that involve fractional exponents requires an understanding of exponent rules, particularly when converting expressions with fractional exponents into radical form or recognizing simplifications when numbers can be expressed as powers of a common base. This concept is critical when evaluating functions with exponents to simplify computations and solve for specific values.

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