Construction A rancher plans to use an existing stone wall and the side of a barn as a

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 112, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1318, use Lagrange multipliers to find the given extremum. In each case, assume that and are positive.

In Exercises 1922, use Lagrange multipliers to find the given extremum of subject to two constraints. In each case, assume that and are nonnegative.

In Exercises 1922, use Lagrange multipliers to find the given extremum of subject to two constraints. In each case, assume that and are nonnegative.

In Exercises 1922, use Lagrange multipliers to find the given extremum of subject to two constraints. In each case, assume that and are nonnegative.

In Exercises 1922, use Lagrange multipliers to find the given extremum of subject to two constraints. In each case, assume that and are nonnegative.

In Exercises 23 and 24, use a spreadsheet to find the given extremum. In each case, assume that and are nonnegative.

In Exercises 23 and 24, use a spreadsheet to find the given extremum. In each case, assume that and are nonnegative.

In Exercises 2528, find three positive numbers and that satisfy the given conditions. The sum is 120 and the product is maximum.

In Exercises 2528, find three positive numbers and that satisfy the given conditions. The sum is 120 and the sum of the squares is minimum.

In Exercises 2528, find three positive numbers and that satisfy the given conditions. The sum is S and the product is maximum.

In Exercises 2528, find three positive numbers and that satisfy the given conditions. The sum is S and the sum of the squares is minimum.

In Exercises 2932, find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.)

In Exercises 2932, find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.)

In Exercises 2932, find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.)

In Exercises 2932, find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.)

Volume Find the dimensions of the rectangular package of largest volume subject to the constraint that the sum of the length and the girth cannot exceed 108 inches (see figure). (Hint: Maximize subject to the constraint )

Cost In redecorating an office, the cost for new carpeting is $3 per square foot and the cost of wallpapering a wall is $1 per square foot. Find the dimensions of the largest office that can be redecorated for $1296 (see figure). (Hint: Maximize subject to

Cost A cargo container (in the shape of a rectangular solid) must have a volume of 480 cubic feet. Use Lagrange multipliers to find the dimensions of the container of this size that has a minimum cost, if the bottom will cost $5 per square foot to construct and the sides and top will cost $3 per square foot to construct.

Cost A manufacturer has an order for 1000 units of fine paper that can be produced at two locations. Let and be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by

Cost A manufacturer has an order for 2000 units of all-terrain vehicle tires that can be produced at two locations. Let and be the numbers of units produced at the two plants. The cost function is modeled by Find the number of units that should be produced at each plant to minimize the cost.

Hardy-Weinberg Law Repeat Exercise 47 in Section 7.5 using Lagrange multipliersthat is, maximize subject to the constraint

Least-Cost Rule The production function for a company is given by where x is the number of units of labor and y is the number of units of capital. Suppose that labor costs $48 per unit, capital costs $36 per unit, and management sets a production goal of 20,000 units. (a) Find the numbers of units of labor and capital needed to meet the production goal while minimizing the cost. (b) Show that the conditions of part (a) are met when This proportion is called the Least-Cost Rule (or Equimarginal Rule).

Least-Cost Rule Repeat Exercise 39 for the production function given by fx, y  100x0.6y0.4.

Production The production function for a company is given by fx, y  100x0.25y0.75 where is the number of units of labor and y is the number of units of capital. Suppose that labor costs $48 per unit and capital costs $36 per unit. The total cost of labor and capital is limited to $100,000. (a) Find the maximum production level for this manufacturer. (b) Find the marginal productivity of money. (c) Use the marginal productivity of money to find the maximum number of units that can be produced if $125,000 is available for labor and capital.

Production Repeat Exercise 41 for the production function given by fx, y  100x0.6y0.4.

Biology A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by where x, y, and z are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions x, y, and z cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

Biology Repeat Exercise 43 for a salt-content model of S  0.01x2y2z2

Animal Shelter An animal shelter buys two different brands of dog food. The number of dogs that can be fed from pounds of the first brand and pounds of the second brand is given by the model Dx, y  x2  52x  y2  44y  256. (a) The shelter orders 100 pounds of dog food. Use Lagrange multipliers to find the number of pounds of each brand of dog food that should be in the order so that the maximum number of dogs can be fed. (b) What is the maximum number of dogs that can be fed?

Nutrition The number of grams of your favorite ice cream can be modeled by Gx, y, z  0.05x2  0.16xy  0.25z2 where is the number of fat grams, is the number of carbohydrate grams, and is the number of protein grams. Use Lagrange multipliers to find the maximum number of grams of ice cream you can eat without consuming more than 400 calories. Assume that there are 9 calories per fat gram, 4 calories per carbohydrate gram, and 4 calories per protein gram.

Construction A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs $10 per foot. To separate the corrals, a fence that costs $4 per foot will divide the region. The total area of the two corrals is to be 6000 square feet. (a) Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing. (b) What is the minimum cost?

Office Space Partitions will be used in an office to form four equal work areas with a total area of 360 square feet (see figure). The partitions that are feet long cost $100 per foot and the partitions that are feet long cost $120 per foot. (a) Use Lagrange multipliers to find the dimensions and that will minimize the cost of the partitions. (b) What is the minimum cost?

Investment Strategy An investor is considering three different stocks in which to invest $300,000. The average annual dividends for the stocks are General Motors (G) 2.7% Pepsico, Inc. (P) 1.7% Sara Lee (S) 2.4%. The amount invested in Pepsico, Inc. must follow the equation 3000S  3000G  P2  0. How much should be invested in each stock to yield a maximum of dividends?

nvestment Strategy An investor is considering three different stocks in which to invest $20,000. The average annual dividends for the stocks are General Mills (G) 2.4% Campbell Soup (C) 1.8% Kellogg Co. (K) 1.9%. The amount invested in Campbell Soup must follow the equation 1000K  1000G  C2  0. How much should be invested in each stock to yield a maximum of dividends?

Advertising A private golf club is determining how to spend its $2700 advertising budget. The club knows from prior experience that the number of responses is given by where is the number of cable television ads, is the number of newspaper ads, and is the number of radio ads. A cable television ad costs $30, a newspaper ad costs $12, and a radio ad costs $15. (a) How much should be spent on each type of advertising to obtain the maximum number of responses? (Assume the golf club uses each type of advertising.) (b) What is the maximum number of responses expected?