Chapter 2

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ 0.5 x^{2}+0.875 x \leq 0.25 $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Endpoints of a diameter are \(P(-1,3)\) and \(Q(7,-5)\)

Find an equation of the perpendicular bisector of the line segment joining the points \(A(1,4)\) and \(B(7,-2)\)

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{3}+11 x \leq 6 x^{2}+6 $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((7,-3) ; \quad\) tangent to the \(x\) -axis

Find the area of the triangle formed by the coordinate axes and the line $$ 2 y+3 x-6=0 $$

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ 16 x^{3}+24 x^{2}>-9 x-1 $$

Plot the points \(M(6,8)\) and \(A(2,3)\) on a coordinate plane. If \(M\) is the midpoint of the line segment \(A B,\) find the coordinates of \(B .\) Write a brief description of the steps you took to find \(B\) and your reasons for taking them.

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Circle lies in the first quadrant, tangent to both \(x\) - and \(y\) -axes; radius 5

(a) Show that if the \(x\) . and \(y\) intercepts of a line are nonzero numbers \(a\) and \(b\) , then the equation of the line can be written in the form $$ \frac{x}{a}+\frac{y}{b}=1 $$ This is called the twe-intercept form of the equation of a line. (b) Use part (a) to find an equation of the line whose \(x\) -intercept is 6 and whose \(y\) -intercept is \(-8\) .

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{1 / 3}

Plot the points \(P(0,3)\) \(Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ \sqrt{0.5 x^{2}+1} \leq 2|x| $$

Grade of a Road West of Albuquerque, New Mexico, Route 40 eastbound is straight and makes a steep descent toward the city. The highway has a 6\(\%\) grade, which means that its slope is \(-\frac{6}{100}\) . Driving on this road, you notice from elevation signs that you have descended a distance of 1000 ft. What is the change in your horizontal distance?

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ (x+1)^{2}<(x-1)^{2} $$

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}-2 x+4 y+1=0 $$

Global Warming Some scientists believe that the average surface temperature of the world has been rising steadily. The average surface temperature can be modeled by $$ T=0.02 t+15.0 $$ where \(T\) is temperature in \(^{\circ} \mathrm{C}\) and \(t\) is years since \(1950 .\) (a) What do the slope and \(T\) -intercept represent? (b) Use the equation to predict the average global surface temperature in 2050 .

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ (x+1)^{2} \leq x^{3} $$

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}-2 x-2 y=2 $$

Drug Dosages If the recommended adult dosage for a drug is \(D\) (in mg), then to determine the appropriate dosage \(c\) for a child of age \(a\) , pharmacists use the equation $$ c=0.0417 D(a+1) $$ Suppose the dosage for an adult is 200 \(\mathrm{mg}\) (a) Find the slope. What does it represent? (b) What is the dosage for a newborn?

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}-4 x+10 y+13=0 $$

Flea Market The manager of a weekend flea market knows from past experience that if she charges \(x\) dollars for a rental space at the flea market, then the number \(y\) of spaces she can rent is given by the equation \(y=200-4 x\) (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) (b) What do the slope, the \(y\) -intercept, and the \(x\) -intercept of the graph represent?

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}+6 y+2=0 $$

Production Cost A small-appliance manufacturer finds that if he produces \(x\) toaster ovens in a month, his production cost is given by the equation $$ y=6 x+3000 $$ (where \(y\) is measured in dollars). (a) Sketch a graph of this linear equation. (b) What do the slope and \(y\) intercept of the graph represent?

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}+x=0 $$

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}+2 x+y+1=0 $$

Crickets and Temperature Biologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at \(70^{\circ} \mathrm{F}\) and 168 chirps per minute at \(80^{\circ} \mathrm{F} .\) (a) Find the linear equation that relates the temperature \(t\) and the number of chirps per minute \(n\) . (b) If the crickets are chirping at 150 chirps per minute, estimate the temperature.

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}-\frac{1}{2} x+\frac{1}{2} y=\frac{1}{8} $$

Depreciation A small business buys a computer for \(\$ 4000\) . After 4 years the value of the computer is expected to be \(\$ 200\) . For accounting purposes the business uses linear depreciation to assess the value of the computer at a given time. This means that if \(V\) is the value of the computer at time \(t\) then a linear equation is used to relate \(V\) and \(t\) (a) Find a linear equation that relates \(V\) and \(t\) (b) Sketch a graph of this linear equation. (c) What do the slope and \(V\) -intercept of the graph represent? (d) Find the depreciated value of the computer 3 years from the date of purchase.

Estimating Profit An appliance manufacturer estimates that the profit \(y\) (in dollars) generated by producing \(x\) cook tops per month is given by the equation $$ y=10 x+0.5 x^{2}-0.001 x^{3}-5000 $$ (a) Graph the equation. (b) How many cooktops must be produced to begin generating a profit? (c) For what range of values of \(x\) is the company's profit greater than \(\$ 15,000 ?\)

\(65-72\) . Show that the equation represents a circle, and find the center and radius of the circle. $$ x^{2}+y^{2}+\frac{1}{2} x+2 y+\frac{1}{16}=0 $$

Pressure and Depth At the surface of the ocean the water pressure is the same as the air pressure above the water, 15 \(\mathrm{lb} / \mathrm{in}^{2}\). Below the surface the water pressure increases by 4.34 \(\mathrm{lb} / \mathrm{in}^{2}\) for every 10 \(\mathrm{ft}\) of descent. (a) Find an equation for the relationship between pressure and depth below the ocean surface. (b) Sketch a graph of this linear equation. (c) What do the slope and \(y\) -intercept of the graph represent? Yd) At what depth is the pressure 100 Ib/in \(^{2} ?\)

\(73-76\) Sketch the graph of the equation. $$ x^{2}+y^{2}+4 x-10 y=21 $$

Distance, Speed, and Time Jason and Debbie leave Detroit at \(2 : 00\) PM. and drive at a constant speed, traveling west on \(1-90\) They pass Ann Arbor, 40 \(\mathrm{mi}\) from Detroit, at \(2:50 \)PM (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?

Misleading Graphs Write a short essay describing different ways in which a graphing calculator might give a misleading graph of an equation.

\(73-76\) Sketch the graph of the equation. $$ 4 x^{2}+4 y^{2}+2 x=0 $$

cost of Driving The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May her driving cost was \(\$ 380\) for 480 \(\mathrm{mi}\) and in June her cost was \(\$ 460\) for 800 \(\mathrm{mi}\) . Assume that there is a linear relationship between the monthly cost \(C\) of driving a car and the distance driven \(d\) (a) Find a linear equation that relates \(C\) and \(d\) (b) Use part (a) to predict the cost of driving 1500 \(\mathrm{mi}\) per month. (c) Draw the graph of the linear equation. What does the slope of the line represent? (d) What does the \(y\) -intercept of the graph represent? (d) Why is a linear relationship a suitable model for this situation?

Algebraic and Graphical Solution Methods Write a short essay comparing the algebraic and graphical methods for solving equations. Make up your own examples to illustrate the advantages and disadvantages of each method.

\(73-76\) Sketch the graph of the equation. $$ x^{2}+y^{2}+6 x-12 y+45=0 $$

Manufacturing Cost The manager of a furniture factory finds that it costs \(\$ 2200\) to manufacture 100 chairs in one day and \(\$ 4800\) to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses this relationship. Then graph the equation. (b) What is the slope of the line in part (a), and what does it represent? (c) What is the \(y\) -intercept of this line, and what does it represent?

\(73-76\) Sketch the graph of the equation. $$ x^{2}+y^{2}-16 x+12 y+200=0 $$

What Does the Slope Mean? Suppose that the graph of the outdoor temperature over a certain period of time is a line. How is the weather changing if the slope of the line is positive? If it is negative? If it is zero?

Enter Equations Carefully A student wishes to graph the equations $$ y=x^{1 / 3} \quad \text { and } \quad y=\frac{x}{x+4} $$ on the same screen, so he enters the following information into his calculator: $$ Y_{1}=x^{\wedge} 1 / 3 \quad Y_{2}=x / x+4 $$ The calculator graphs two lines instead of the equations he wanted. What went wrong?

\(77-82\) . Test the equation for symmetry. $$ y=x^{4}+x^{2} $$

Collinear Points Suppose you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?

\(77-82\) . Test the equation for symmetry. $$ x=y^{4}-y^{2} $$

\(77-82\) . Test the equation for symmetry. $$ y=x^{3}+10 x $$

\(77-82\) . Test the equation for symmetry. $$ y=x^{2}+|x| $$

\(77-82\) . Test the equation for symmetry. $$ x^{4} y^{4}+x^{2} y^{2}=1 $$

\(77-82\) . Test the equation for symmetry. $$ x^{2} y^{2}+x y=1 $$