   Chapter 14.5, Problem 10E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Find the maximum value of z =   x y 2 subject to 2 x 2 + y 2 =   600 ;   x   ≥   0 ,   y   ≥   0 .

To determine

To calculate: The maximum value of z=xy2 subject to 2x2+y2=600, x0, y0.

Explanation

Given Information:

The provided function is z=xy2 subject to 2x2+y2=600, x0, y0.

Formula used:

According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0,

(1) Find the critical values of f(x,y) using the new variable λ to form the objective function F(x,y,λ)=f(x,y)+λg(x,y).

(2) The critical points of f(x,y) are the critical values of F(x,y,λ) which satisfies g(x,y)=0.

(3) The critical points of F(x,y,λ) are the points that satisfy Fx=0, Fy=0, and Fλ=0, that is, the points which make all the partial derivatives of zero.

For a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by fy.

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0.

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

Consider the function, z=xy2.

The provided constraint is 2x2+y2=600, x0, y0.

According to the Lagrange multipliers method,

The objective function is F(x,y,λ)=f(x,y)+λg(x,y).

Thus, f(x,y)=xy2 and g(x,y)=2x2+y2600.

Substitute xy2 for f(x,y) and 2x2+y2=600 for g(x,y) in F(x,y,λ)=f(x,y)+λg(x,y).

F(x,y,λ)=xy2+λ(2x2+y2600)

Since, the critical points of F(x,y,λ) are the points that satisfy Fx=0, Fy=0, and Fλ=0

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 516, evaluate the given quantity. log100,000

Finite Mathematics and Applied Calculus (MindTap Course List)

True or False: Du f(a, b) = ∇(a, b) · u.

Study Guide for Stewart's Multivariable Calculus, 8th

Using as an estimator, find the maximum error in estimating with the Midpoint Rule and 8 subintervals. 9

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 