Chapter 13

A manufacturer has an order for 2000 units of all-terrain vehicle tires that can be produced at two locations. Let \(x_{1}\) and \(x_{2}\) be the numbers of units produced at the two plants. The cost function is modeled by \(C=0.25 x_{1}^{2}+10 x_{1}+0.15 x_{2}^{2}+12 x_{2}\) Find the number of units that should be produced at each plant to minimize the cost.

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from \(x_{1}\) units of running shoes and \(x_{2}\) units of basketball shoes is \(R=-5 x_{1}^{2}-8 x_{2}^{2}-2 x_{1} x_{2}+42 x_{1}+102 x_{2}\) where \(x_{1}\) and \(x_{2}\) are in thousands of units. Find \(x_{1}\) and \(x_{2}\) so as to maximize the revenue.

Evaluate \(w_{x}, w_{y}\), and \(w_{z}\) at the point. $$ w=2 x z^{2}+3 x y z-6 y^{2} z \quad(1,-1,2) $$

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=x y \quad c=\pm 1, \pm 2, \ldots, \pm 6 $$

Describe the traces of the surface in the given planes. $$ x^{2}-y-z^{2}=0 \quad x y \text { -plane, } y=1, y z \text { -plane } $$

Find the standard equation of the sphere. Endpoints of a diameter: \((2,0,0),(0,6,0)\)

Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ (0,0.5),(1,7.6),(3,60),(4.2,117),(5,170),(7.9,380) $$

Repeat Exercise 47 in Section \(13.5\) using Lagrange multipliers - that is, maximize \(P(p, q, r)=2 p q+2 p r+2 q r\) subject to the constraint \(p+q+r=1\)

A retail outlet sells two types of riding lawn mowers, the prices of which are \(p_{1}\) and \(p_{2}\). Find \(p_{1}\) and \(p_{2}\) so as to maximize total revenue, where \(R=515 p_{1}+805 p_{2}+1.5 p_{1} p_{2}-1.5 p_{1}^{2}-p_{2}^{2}\)

Evaluate \(w_{x}, w_{y}\), and \(w_{z}\) at the point. $$ w=x y e^{z^{2}} \quad(2,1,0) $$

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=e^{x y} \quad c=1,2,3,4, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} $$

Describe the traces of the surface in the given planes. $$ \begin{aligned} &y=x^{2}+z^{2}\\\ &x y \text { -plane, } y=1, y z \text { -plane } \end{aligned} $$

Find the standard equation of the sphere. Endpoints of a diameter: \((1,0,0),(0,5,0)\)

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,4),(2,6),(3,8),(4,11),(5,13),(6,15) $$

The production function for a company is given by \(f(x, y)=100 x^{0.25} y^{0.75}\) 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 \(\frac{\text { Marginal productivity of labor }}{\text { Marginal productivity of capital }}=\frac{\text { unit price of labor }}{\text { unit price of capital }}\) This proportion is called the Least-Cost Rule (or Equimarginal Rule).

Find \(p_{1}\) and \(p_{2}\) so as to maximize the total revenue \(R=x_{1} p_{1}+x_{2} p_{2}\) for a retail outlet that sells two competitive products with the given demand functions. $$ x_{1}=1000-2 p_{1}+p_{2}, x_{2}=1500+2 p_{1}-1.5 p_{2} $$

Find values of \(x\) and \(y\) such that \(f_{x}(x, y)=0\) and \(f_{y}(x, y)=0\) simultaneously. $$ f(x, y)=x^{2}+4 x y+y^{2}-4 x+16 y+3 $$

Describe the traces of the surface in the given planes. $$ \frac{x^{2}}{4}+y^{2}+z^{2}=1 \quad x y \text { -plane, } x z \text { -plane, } y z \text { -plane } $$

Find the standard equation of the sphere. Center: \((-2,1,1) ;\) tangent to the \(x y\) -plane

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,7.5),(2,7),(3,7),(4,6),(5,5),(6,4.9) $$

Find \(p_{1}\) and \(p_{2}\) so as to maximize the total revenue \(R=x_{1} p_{1}+x_{2} p_{2}\) for a retail outlet that sells two competitive products with the given demand functions. $$ x_{1}=1000-4 p_{1}+2 p_{2}, x_{2}=900+4 p_{1}-3 p_{2} $$

Find values of \(x\) and \(y\) such that \(f_{x}(x, y)=0\) and \(f_{y}(x, y)=0\) simultaneously. $$ f(x, y)=3 x^{3}-12 x y+y^{3} $$

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=\ln (x-y) \quad c=0, \pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 2 $$

Describe the traces of the surface in the given planes. $$ y^{2}+z^{2}-x^{2}=1 \quad x y \text { -plane, } x z \text { -plane, } y z \text { -plane } $$

Find the standard equation of the sphere. Center: \((1,2,0) ;\) tangent to the \(y z\) -plane

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=9-x^{2}, y=0 $$

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,3),(2,6),(3,2),(4,3),(5,9),(6,1) $$

The production function for a company is given by \(f(x, y)=100 x^{0.25} y^{0.75}\) 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 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.

A corporation manufactures a high-performance automobile engine product at two locations. The cost of producing \(x_{1}\) units at location 1 is \(C_{1}=0.05 x_{1}^{2}+15 x_{1}+5400\) and the cost of producing \(x_{2}\) units at location 2 is \(C_{2}=0.03 x_{2}^{2}+15 x_{2}+6100\) The demand function for the product is \(p=225-0.4\left(x_{1}+x_{2}\right)\) and the total revenue function is \(R=\left[225-0.4\left(x_{1}+x_{2}\right)\right]\left(x_{1}+x_{2}\right)\) Find the production levels at the two locations that will maximize the profit \(P=R-C_{1}-C_{2}\)

A manufacturer estimates the Cobb-Douglas production function to be given by \(f(x, y)=100 x^{0.75} y^{0.25}\) Estimate the production levels when \(x=1500\) and \(y=1000\)

Identify the quadric surface. $$ x^{2}+\frac{y^{2}}{4}+z^{2}=1 $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x^{3 / 2}, y=x $$

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (0.5,2),(0.75,1.75),(1,3),(1.5,3.2),(2,3.7),(2.6,4) $$

Repeat Exercise 41 for the production function given by \(f(x, y)=100 x^{0.6} y^{0.4}\)

A corporation manufactures candles at two locations. The cost of producing \(x_{1}\) units at location 1 is \(C_{1}=0.02 x_{1}^{2}+4 x_{1}+500\) and the cost of producing \(x_{2}\) units at location 2 is \(C_{2}=0.05 x_{2}^{2}+4 x_{2}+275 .\) The candles sell for \(\$ 15\) per unit. Find the quantity that should be produced at each location to maximize the profit \(P=15\left(x_{1}+x_{2}\right)-C_{1}-C_{2}\)

Identify the quadric surface. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1 $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ 2 x-3 y=0, x+y=5, y=0 $$

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (1,36),(2,10),(3,0),(4,4),(5,16),(6,36) $$

Find the dimensions of a rectangular package of maximum volume that may be sent by a shipping company assuming that the sum of the length and the girth (perimeter of a cross section) cannot exceed 96 inches.

Find the slope of the surface at the given point in (a) the \(x\) -direction and (b) the \(y\) -direction. $$ \begin{aligned} &z=x y \\ &(1,2,2) \end{aligned} $$

A sporting goods manufacturer produces regulation soccer balls at two plants. The costs of producing \(x_{1}\) units at location 1 and \(x_{2}\) units at location 2 are given by $$ \begin{aligned} &C_{1}\left(x_{1}\right)=0.02 x_{1}^{2}+4 x_{1}+500\\\ &\text { and }\\\ &C_{2}\left(x_{2}\right)=0.05 x_{2}^{2}+4 x_{2}+275 \end{aligned} $$ respectively. If the product sells for \(\$ 50\) per unit, then the profit function for the product is given by $$ \begin{aligned} &P\left(x_{1}, x_{2}\right)=50\left(x_{1}+x_{2}\right)-C_{1}\left(x_{1}\right)-C_{2}\left(x_{2}\right)\\\ &\text { Find (a) } P(250,150) \text { and (b) } P(300,200) \text { . } \end{aligned} $

Identify the quadric surface. $$ 25 x^{2}+25 y^{2}-z^{2}=5 $$

Find the sphere's center and radius. $$ x^{2}+y^{2}+z^{2}-2 x+6 y+8 z+1=0 $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ x y=9, y=x, y=0, x=9 $$

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (0.5,9),(1,8.5),(1.5,7),(2,5.5),(2.5,5),(3,3.5) $$

The average amount of time that a customer waits in line for service is given by \(W(x, y)=\frac{1}{x-y}, \quad y

Identify the quadric surface. $$ 9 x^{2}+4 y^{2}-8 z^{2}=72 $$

Find the sphere's center and radius. $$ x^{2}+y^{2}+z^{2}-4 y+6 z+4=0 $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x, y=2 x, x=2 $$

An animal shelter buys two different brands of dog food. The number of dogs that can be fed from \(x\) pounds of the first brand and \(y\) pounds of the second brand is given by the model \(D(x, y)=-x^{2}+52 x-y^{2}+44 y+256\) (a) The shelter orders 100 pounds of \(\operatorname{dog}\) 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?