Chapter 13

How many real solutions are possible for a system of equations whose graphs are an ellipse and a line?

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{(x-1)^{2}}{49}+\frac{(y+2)^{2}}{25}=1 $$

Write an equation of the circle with the given center and radius. See Example 8. $$ (0,0) ; \sqrt{3} $$

Discuss how graphing a linear inequality such as \(x+y<9\) is similar to graphing a nonlinear inequality such as \(x^{2}+y^{2}<9\)

Solve. The sum of the squares of two numbers is 130 . The difference of the squares of the two numbers is 32 . Find the two numbers.

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ y^{2}=x^{2}+16 $$

Write an equation of the circle with the given center and radius. See Example 8. $$ (0,-6) ; \sqrt{2} $$

Discuss how graphing a linear inequality such as \(x+y<9\) is different from graphing a nonlinear inequality such as \(x^{2}+y^{2}<9\)

Solve. The sum of the squares of two numbers is \(20 .\) Their product is 8 . Find the two numbers.

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \left(x+\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}=1 $$

Write an equation of the circle with the given center and radius. See Example 8. $$ (-5,4) ; 3 \sqrt{5} $$

$$ \begin{aligned} &\text { Graph the system: }\\\ &\left\\{\begin{array}{l} y \leq x^{2} \\ y \geq x+2 \\ x \geq 0 \\ y \geq 0 \end{array}\right. \end{aligned} $$

Solve. During the development stage of a new rectangular keypad for a security system, it was decided that the area of the rectangle should be 285 square centimeters and the perimeter should be 68 centimeters. Find the dimensions of the keypad.

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts.] $$ y=-2 x^{2}+4 x-3 $$

Write an equation of the circle with the given center and radius. See Example 8. The origin; \(4 \sqrt{7}\)

$$ \begin{aligned} &\text { Graph the system: }\\\ &\left\\{\begin{array}{l} x \geq 0 \\ y \geq 0 \\ y \geq x^{2}+1 \\ y \leq 4-x \end{array}\right. \end{aligned} $$

Solve. A rectangular holding pen for cattle is to be designed so that its perimeter is 92 feet and its area is 525 feet. Find the dimensions of the holding pen.

Perform each indicated operation. $$ \left(2 x^{3}\right)\left(-4 x^{2}\right) $$

Graph each equation. See Sections 3.2 and 3.3. $$ y=2 x+5 $$

Recall that in business, a demand function expresses the quantity of a commodity demanded as a function of the commodity's unit price. A supply function expresses the quantity of a commodity supplied as a function of the commodity's unit price. When the quantity produced and supplied is equal to the quantity demanded, then we have what is called market equilibrium. Use this information for Exercises 45 and \(46 .\) The demand function for a certain compact disc is given by the function \(p(x)=-0.01 x^{2}-0.2 x+9\) and the corresponding supply function is given by \(p(x)=0.01 x^{2}-0.1 x+3,\) where \(p(x)\) is in dollars and \(x\) is in thousands of units. Find the equilibrium quantity and the corresponding price by solving the system consisting of the two given equations.

Perform each indicated operation. $$ 2 x^{3}-4 x^{3} $$

Graph each equation. See Sections 3.2 and 3.3. $$ y=-3 x+3 $$

Recall that in business, a demand function expresses the quantity of a commodity demanded as a function of the commodity's unit price. A supply function expresses the quantity of a commodity supplied as a function of the commodity's unit price. When the quantity produced and supplied is equal to the quantity demanded, then we have what is called market equilibrium. Use this information for Exercises 45 and \(46 .\) The demand function for a certain style of picture frame is given by the function \(p(x)=-2 x^{2}+90\) and the corresponding supply function is given by \(p(x)=9 x+34,\) where \(p(x)\) is in dollars and \(x\) is in thousands of units. Find the equilibrium quantity and the corresponding price by solving the system consisting of the two given equations.

Perform each indicated operation. $$ -5 x^{2}+x^{2} $$

Graph each equation. See Sections 3.2 and 3.3. $$ y=3 $$

Perform each indicated operation. $$ \left(-5 x^{2}\right)\left(x^{2}\right) $$

Graph each equation. See Sections 3.2 and 3.3. $$ x=-2 $$

The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercepts. How much longer? $$ \frac{x^{2}}{16}+\frac{y^{2}}{25}=1 $$

Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{1}{\sqrt{3}} $$

The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercepts. How much longer? $$ \frac{x^{2}}{100}+\frac{y^{2}}{49}=1 $$

Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{\sqrt{5}}{\sqrt{8}} $$

The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercepts. How much longer? $$ 4 x^{2}+y^{2}=16 $$

Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{4 \sqrt{7}}{\sqrt{6}} $$

The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercepts. How much longer? $$ x^{2}+4 y^{2}=36 $$

Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{10}{\sqrt{5}} $$

If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their equations.

We know that \(x^{2}+y^{2}=25\) is the equation of a circle. Rewrite the equation so that the right side is equal to \(1 .\) Which type of conic section does this equation form resemble? In fact, the circle is a special case of this type of conic section. Describe the conditions under which this type of conic section is a circle.

Although there are many larger observation wheels on the horizon, as of this writing the largest observation wheel in the world is the Singapore Flyer. From the Flyer, you can see up to \(45 \mathrm{~km}\) away. Each of the 28 enclosed capsules holds 28 passengers, completes a full rotation every 32 minutes. Its diameter is 150 meters, and the height of this giant wheel is 165 meters. (Source: singaporeflyer.com) a. What is the radius of the Singapore Flyer? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the Singapore Flyer.

In 1893, Pittsburgh bridge builder George Ferris designed and built a gigantic revolving steel wheel whose height was 264 feet and diameter was 250 feet. This Ferris wheel opened at the 1893 exposition in Chicago. It had 36 wooden cars, each capable of holding 60 passengers. (Source: The Handy Science Answer Book) a. What was the radius of this Ferris wheel? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the wheel.

The world's largest-diameter Ferris wheel currently operating is the Cosmo Clock 21 at Yokohama City, Japan. It has a 60 -armed wheel, its diameter is 100 meters, and it has a height of 105 meters. (Source: The Handy Science Answer Book) a. What is the radius of this Ferris wheel? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the wheel.

If you are given a list of equations of circles and parabolas and none are in standard form, explain how you would determine which is an equation of a circle and which is an equation of a parabola. Explain also how you would distinguish the upward or downward parabolas from the left-opening or right- opening parabolas.

Determine whether the triangle with vertices (2,6) \((0,-2),\) and (5,1) is an isosceles triangle.

Solve. Two surveyors need to find the distance across a lake. They place a reference pole at point \(A\) in the diagram. Point \(B\) is 3 meters east and 1 meter north of the reference point \(A .\) Point \(C\) is 19 meters east and 13 meters north of point \(A\). Find the distance across the lake, from \(B\) to \(C\).

Solve. A bridge constructed over a bayou has a supporting arch in the shape of a parabola. Find an equation of the parabolic arch if the length of the road over the arch is 100 meters and the maximum height of the arch is 40 meters.

Solve. Cindy Brown, an architect, is drawing plans on grid paper for a circular pool with a fountain in the middle. The paper is marked off in centimeters, and each centimeter represents 1 foot. On the paper, the diameter of the "pool" is 20 centimeters, and "fountain" is the point (0,0) . a. Sketch the architect's drawing. Be sure to label the axes. b. Write an equation that describes the circular pool. c. Cindy plans to place a circle of lights around the fountain such that each light is 5 feet from the fountain. Write an equation for the circle of lights and sketch the circle on your drawing.