Optimization Problems (10 problems)

A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be 6,400 yd^2 and no fencing is needed along the river bank. Find the dimensions of the pasture that will require the least amount of fencing.

If x thousand dollars is spent on labor and y thousand dollars is spent on equipment, the output at a certain factory will beunits. If $120,000 is available, how should this be allocated between labor and equipment to generate the largest possible output?

Find the maximum and minimum values of the function subject to the constraint

Use Lagrange Multiplier to minimize

subject to .

Find the minimum value of subject to

Let. Find the minimum value of the function f(x, y) subject to the constraint

Find the minimum value of the function subject to the constraint

There are 320 meters of fencing available to enclose a rectangular field. How should the fencing be used so that the enclosed area is as large as possible?

Given a positive number a, decompose it into a sum of n numbers whose squares add up to the minimal possible number

Find the extrema values ofon the ellipse

using Lagrange multipliers.