Exam 8: Analyzing Multivariable Change: Optimization

Consider the following contour graph and three-dimensional graph for a function F with inputs x and y. On the basis of the graphs approximate the inputs and output of each critical point in the form

A fitness center is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and in the newspaper. Previous experience with such advertising leads the manager to expect responses when r adds are run on the radio and n adds appear in the newspaper. Each ad on the radio costs $13 and each newspaper ad costs $3. Using Lagrange multipliers, a student found that when $504 are budgeted for advertising, the maximum number of responses occurred from 26 radio ad and 56 newspaper ads and had Suppose the manager budgeted an additional $22 for advertising. What is the approximate change in the optimal value as a result of this change in the constraint level?

Express SSE as a multivariable function of a and b for the best fitting line Use the data below.

The daily output at a plant manufacturing transistor radios is approximated by the function radios, where L is the size of the labor force measured in hundreds of worker hours and K is the capital investment in thousands of dollars. Suppose the plant manager has a daily budget of $18,000 to be invested in capital or spent on labor and that the average wage of an employee at the radio plant is $7.00 per hour. What combination of worker hours and capital expenditures will yield the maximum daily production? Round your answers to the nearest integer.

The figures show a contour graph for a function f in blue with a constraint function in black. Locate any optimal points of and classify each as a relative maximum point or a relative minimum point.

Test for relative maxima and minima.

A restaurant mixes ground beef that costs $b per pound with pork sausage that costs $p per pound to make a meat mixture that is used on the restaurant's signature pizza. The quarterly revenue, in thousands of dollars, from the sale of this pizza is given by the equation To the nearest thousand dollars, what is the maximum quarterly revenue from the sale of the restaurant's signature pizza?

Express SSE as a multivariable function of a and b for the best fitting line Use the data below.

For a region, the percentage of adults 20-24 years of age how have not been married is given in the table below. The percentage of adults 20-24 years old who had never been married in 1960 was 37. Add this data point and use technology to find a linear model. What is the percentage of adults 20-24 years old in 2000 who have never been married, according to this model? Round your answer to the nearest integer.

A process to extract pigment from sunflower seeds involves washing the sunflower heads in heated water. Suppose the percentage of pigment that can be removed from the sunflower head by washing for 20 minutes is percent, where r is the amount of milliliters of water per gram of sunflower used for washing and is the water temperature. Find the critical point of the pigment removal function. Round your answers to four decimal places.

Locate and classify any critical points.

A model for the elevation above sea level of a tract of farmland is feet above sea level, where e is the distance in miles east of the western fence and n is the distance in miles north of the southern fence. One critical point is Use the Determinant Test to classify it as a maximum, minimum, or saddle point.

Locate and classify any critical points.

Milk proteins are sometimes added to sausage to reduce shrinking due to cooking loss and improve the texture of the sausage. Suppose the cooking loss (expressed as a percentage of initial weight) can be modeled by percent, where w is the proportion of whey protein and s is the proportion of skim milk powder. Determine whether the function L has a relative maximum, relative minimum, or a saddle point, and find the corresponding proportions of whey protein and skim milk powder at that point. Round your answer to the nearest hundredth of a percent.

A model for the elevation above sea level of a tract of farmland is feet above sea level, where e is the distance in miles east of the western fence and n is the distance in miles north of the southern fence. Find the two critical points of E. Round your answers to three decimal places.

A nursery sells mulch by the truckload. Bark mulch sells for $b per load and pine straw sells for $p per load. The nursery's average weekly profit form the sale of these two types of mulch can be modeled by the equation dollars. To the nearest dollar, what is the maximum weekly profit from the sales of these two types of mulch?

A rancher removed 200 feet of wire fencing from a field on his ranch. He wants to reuse the fencing to create a rectangular corral into which he will build a 6-foot-wide wooden gate. The dimensions of the corral with the greatest possible area are found using the multivariable functions for the amount of fencing and for the resulting area of the corral: feet is the amount of fencing needed for the specified rectangular corral of width w feet and length l feet. The area of the specified corral is where w feet is the width and l feet is the length. Write the multivariable function to be maximized.

Is the point and on the contour graph below a relative maximum point, a relative minimum point, or a saddle point?

Test for relative maxima and minima.

The table below gives the number of inches of precipitation that fell in a city in the given months. Use the method of least squares to find the multivariable function f with inputs a and b for the best fitting line where x is 1 in January, 2 in February, and 3 in March.