Chapter 1

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |-2| ______ |-5|

In a simple model of the economy (by John Maynard Keynes), equilibrium between national output and national expenditures is given by the equilibrium equation $$ Y=C+I+G+(X-M) $$ where \(Y\) is the national income, C is consumption (which depends on national income), I is the amount of investment, \(G\) is government spending, \(X\) is exports, and \(M\) is imports (which also depend on national income). Solve the equilibrium equation for \(Y\) under the given conditions. $$\begin{aligned} &C=120+.9 Y, M=20+.2 Y, I=140, G=150, \text { and }\\\ &X=60 \end{aligned}$$

The number of unmarried couples in the United States who live together was 3.2 million in 1990 and grew in a linear fashion to 5.5 million in 2000 . (a) Let \(x=0\) correspond to \(1990 .\) Write a linear equation expressing the number \(y\) of unmarried couples living together (in millions) in year \(x\). (b) Assuming the equation remains accurate, estimate the number of unmarried couples living together in 2010 . (c) When will the number of unmarried couples living together reach \(10,100,000 ?\)

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. 5 ________ |-2|

In a simple model of the economy (by John Maynard Keynes), equilibrium between national output and national expenditures is given by the equilibrium equation $$ Y=C+I+G+(X-M) $$ where \(Y\) is the national income, C is consumption (which depends on national income), I is the amount of investment, \(G\) is government spending, \(X\) is exports, and \(M\) is imports (which also depend on national income). Solve the equilibrium equation for \(Y\) under the given conditions. $$\begin{aligned} &C=60+.85 Y, M=35+.2 Y, I=95, G=145, \text { and }\\\ &X=50 \end{aligned}$$

The percentage of people 25 years old and older who have a Bachelor's degree or higher was about 25.6 in 2000 and 27.7 in 2004 (a) Find a linear equation that gives the percentage of people 25 and over who have a Bachelor's degree or higher in terms of time \(t\), where \(t\) is the number of years since 2000 . Assume that this equations remains valid in the future. (b) What will the percentage be in \(2010 ?\) (c) When will \(34 \%\) of those 25 and over have a Bachelor's degree or higher?

Do Exercise 80 with \((c, d)\) and \((c, k)\) in place of (2,1) and (2,5).

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |3| __________-|4|

Use the height equation in Example 12 . Note that an object that is dropped (rather than thrown downward has initial velocity \(v_{0}=0\). How long does it take a baseball to reach the ground if it is dropped from the top of a 640 -foot-high building? Compare with Example 12 .

At the Factory in Example \(14,\) the cost of producing \(x\) can openers is given by \(y=2.75 x+26,000\) (a) Write an equation that gives the average cost per can opener when \(x\) can openers are produced. (b) How many can openers should be made to have an average cost of \(\$ 3\) per can opener?

Find the three points that divide the line segment from (-4,7) to (10,-9) into four parts of equal length.

Fill the blank with \(<,=\), or \(>\) so that the resulting statement is true. |-3| ________ 0

Use the height equation in Example 12 . Note that an object that is dropped (rather than thrown downward has initial velocity \(v_{0}=0\). You are standing on a cliff that is 200 feet high. How long will it take a rock to reach the ground if (a) you drop it? (b) you throw it downward at an initial velocity of 40 feet per second? (c) How far does the rock fall in 2 seconds if you throw it downward with an initial velocity of 40 feet per second?

Suppose the cost of making \(x\) TV sets is given by \(y=145 x+120,000\) (a) Write an equation that gives the average cost per set when \(x\) sets are made. (b) How many sets should be made in order to have an average cost per set of \(\$ 175 ?\)

Find all points \(P\) on the \(x\) -axis that are 5 units from (3,4) [Hint: \(P\) must have coordinates \((x, 0)\) for some \(x\) and the distance from \(P \text { to }(3,4) \text { is } 5 .]\)

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. -7= _______ |-1|

Use the height equation in Example 12 . Note that an object that is dropped (rather than thrown downward has initial velocity \(v_{0}=0\). A rocket is fired straight up from ground level with an initial velocity of 800 feet per second. (a) How long does it take the rocket to rise 3200 feet? (b) When will the rocket hit the ground?

The profit \(p\) (in thousands of dollars) on \(x\) thousand units of a specialty item is \(p=.6 x-14.5 .\) The cost \(c\) of manufacturing \(x\) thousand items is given by \(c=.8 x+14.5\). (a) Find an equation that gives the revenue \(r\) from selling \(x\) thousand items. (b) How many items must be sold for the company to break even (i.e., for revenue to equal cost)?

Find all points on the \(y\) -axis that are 8 units from (-2,4).

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. -|-4| _________ 0

A rocket loaded with fireworks is to be shot vertically upward from ground level with an initial velocity of 200 feet per second. When the rocket reaches a height of 400 feet on its upward trip the fireworks will be detonated. How many seconds after liftoff will this take place?

A publisher has fixed costs of \(\$ 110,000\) for a mathematics text. The variable costs are \(\$ 50\) per book. The book sells for \(\$ 72 .\) Find equations that give, (a) The cost \(c\) of making \(x\) books (b) The revenue \(r\) from selling \(x\) books (c) The profit \(p\) from selling \(x\) books (d) What is the publisher's break-even point (see Exer cise \(83(b)) ?\)

Find the distance between the given numbers. -3 and 4

The atmospheric pressure \(a\) (in pounds per square foot) at height \(h\) thousand feet above sea level is approximately $$ a=8315 h^{2}-73.93 h+2116.1 $$ (a) Find the atmospheric pressure at sea level and at the top of Mount Everest, the tallest mountain in the world \(\left(29,035 \text { feet }^{*}\right) .\) [Remember that \(h\) is measured in thousands.] (b) The atmospheric pressure at the top of Mount Rainier is 1223.43 pounds per square foot. How high is Mount Rainier?

Find the distance between the given numbers. 7 and 107

Data from the U.S. Department of Health and Human Services indicates that the cumulative number \(N\) of reported cases of AIDS in the United States in year \(x\) can be approximated by the equation $$ N=3362.1 x^{2}-17,270.3 x+24,043 $$ where \(x=0\) corresponds to \(1980 .\) In what year did the total reach \(550,000 ?\)

Find a number \(x\) such that \((0,0),(3,2),\) and \((x, 0)\) are the vertices of an isosceles triangle, neither of whose two equal sides lie on the \(x\) -axis.

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. -7 and \(15 / 2\)

According to data from the U.S. Census Bureau, the population \(P\) of Cleveland, Ohio (in thousands) in year \(x\) can be approximated by \(P=.08 x^{2}-13.08 x+927,\) where \(x=0\) corresponds to \(1950 .\) In what year in the past was the population about \(804,200 ?\)

Use the graph and the following information for Exercises \(85-86 .\) Rocky is an "independent" ticket dealer who markets choice tickets for Los Angeles Lakers home games. (California currently has no laws against ticket scalping.) Each graph shows how many tickets will be demanded by buyers at a particular price. For instance, when the Lakers play the Chicago Bulls, the graph shows that at a price of \(\$ 160,\) no tickets are demanded. As the price (y-coordinate) gets lower, the number of tickets demanded (x-coordinate) increases.(GRAPH CAN'T COPY) The Fahrenheit and Celsius scales for measuring temperatures are linearly related. They are calibrated using the freezing and boiling points of water at sea level.$$\begin{array}{|l|c|c|}\hline \begin{array}{l}\text { Temperature } \\\\\text { Scale }\end{array} & \begin{array}{c}\text { Fahrenheit } \\ \text { Scale }\end{array} & \begin{array}{c}\text { Celsius } \\\\\text { Scale }\end{array} \\\\\hline \text { Water Freezes } & 32^{\circ} & 0^{\circ} \\\\\hline \text { Water Boils } & 212^{\circ} & 100^{\circ} \\ \hline\end{array}$$.(a) Use the data in the table to write a formula that relates the Fahrenheit temperature \(F\) to the Celsius temperature \(C .\) Your answer should be in the form \(F=m C+b\) (b) Solve the equation in part (a) for \(C\) to find a formula that relates the Celsius temperature to the Fahrenheit temperature. (c) When is the temperature in degrees Fahrenheit the same as the temperature in degrees Celsius?

The number \(N\) of AIDS cases diagnosed to date (in thousands) is approximated by \(N=-.37 x^{2}+59.5 x+247.26\) where \(x\) is the number of years since \(1990 .^{\dagger}\) Assuming that this equation remains valid through 2011 , determine when the number of diagnosed cases of AIDS was or will be (a) 825,000 (b) 1.2 million

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. \(\pi\) and 3

The number \(N\) of Walgreens drugstores in year \(x\) can be approximated by \(N=6.82 x^{2}-1.55 x+666.8,\) where \(x=0\) corresponds to \(1980 .^{*}\) Determine when the number of stores was or will be (a) 4240 (b) 5600 (c) 7000

Find the distance between the given numbers. \(\pi\) and -3

The total resources \(T\) (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions, can be approximated by the equation \(T=-.26 x^{2}+3.62 x+30.18,\) where \(x\) is the number of years after \(2000 .^{\dagger}\) Determine when the total resources are at the given level. (a) \(\$ 42.5\) billion (b) \(\$ 30\) billion (c) When will the Corporation be out of money \((T=0) ?\)

The poverty level income for a family of four was \(\$ 13,359\) in \(1990 .\) Because of inflation and other factors, the poverty level rose approximately linearly to \(\$ 19,307\) in 2004 . (a) At what rate is the poverty level increasing? (b) Estimate the poverty level in 2000 and 2009 .

Find the distance between the given numbers. \(\sqrt{2}\) and \(\sqrt{3}\)

Show that the diagonals of a rectangle have the same length. [Hint: Place the rectangle in the first quadrant of the plane and label its vertices appropriately, as in Exercises \(89-90 .\) ]

According to data from the National Highway Traffic Safety Administration, the driver fatality rate \(D\) per 1000 licensed drivers every 100 million miles can be approximated by the equation \(D=.0031 x^{2}-.291 x+7.1,\) where \(x\) is the age of the driver. (a) For what ages is the driver fatality rate about 1 death per \(1000 ?\) (b) For what ages is the rate three times greater than in part (a)?

A Honda Civic LX sedan is worth \(\$ 15,350 now and will be worth \)\$ 9910\( in four years. (a) Assuming linear depreciation, find the cquation that gives the value \)y\( of the car in year \)x$ (b) At what rate is the car depreciating? (c) Estimate the value of the car six years from now.

Find the distance between the given numbers. \(\pi\) and \(\sqrt{2}\)

If the diagonals of a parallelogram have the same length, show that the parallelogram is actually a rectangle. [Hint: See Exercise 90.1

A house in Shaker Heights, Ohio was bought for \(\$ 160,000\) in \(1980 .\) It increased in value in an approximately linear fashion and sold for \(\$ 359,750\) in 1997 . (a) At what rate did the house appreciate (increase in value) during this period? (b) If this appreciation rate remained accurate what would the house be worth in \(2010 ?\)

Galileo discovered that the period of a pendulum depends only on the length of the pendulum and the acceleration of gravity. The period \(T\) of a pendulum (in seconds) is $$T=2 \pi \sqrt{\frac{l}{g}}$$ where \(l\) is the length of the pendulum in feet and \(g \approx\) 32.2 \(\mathrm{ft} / \mathrm{sec}^{2}\) is the acceleration due to gravity. Find the period of a pendulum whose length is 4 feet.

For each nonzero real number \(k\), the graph of \((x-k)^{2}+y^{2}=k^{2}\) is a circle. Describe all possible such circles.

The cost-benefit equation \(\frac{18 x}{100-x}=D\) relates the cost \(D\) (in thousands of dollars) needed to remove \(x\) percent of a polIutant from the emissions of a factory. Find the percent of the pollutant removed when the following amounts are spent. (a) \(\$ 50,000[\text { Here } D=50]\) (b) \(\$ 100,000\) (c) \(\$ 200,000\)

Show that two nonvertical lines with the same slope are parallel. [Hint: The equations of distinct lines with the same slope must be of the form \(y=m x+b\) and \(y=m x+c\) with \(b \neq c\) (why?). If \(\left(x_{1}, y_{1}\right)\) were a point on both lines, its coordinates would satisfy both equations. Show that this leads to a contradiction, and conclude that the lines have no point in common.]

Suppose every point in the coordinate plane is moved 5 units straight up. (a) To what point does each of these points \(\operatorname{go:}(0,-5)\) \((2,2),(5,0),(5,5),(4,1) ?\) (b) Which points go to each of the points in part (a)? (c) To what point does \((a, b)\) go? (d) To what point does \((a, b-5)\) go? (e) What point goes to \((-4 a, b) ?\) (f) What points go to themselves?

(a) Let \(b\) be a real number. Multiply out the expression \(\left(x+\frac{b}{2}\right)^{2}\) (b) Explain why your computation in part (a) shows that this statement is true: If you add \(\left(\frac{b}{2}\right)^{2}\) to the expression \(x^{2}+b x,\) the resulting polynomial is a perfect square.

Prove that nonvertical parallel lines \(L\) and \(M\) have the same slope, as follows. Suppose \(M\) lies above \(L\), and choose two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) on \(L\). (a) Let \(P\) be the point on \(M\) with first coordinate \(x_{1}\). Let \(b\) denote the vertical distance from \(P\) to \(\left(x_{1}, y_{1}\right) .\) Show that the second coordinate of \(P\) is \(y_{1}+b\) (b) Let \(Q\) be the point on \(M\) with first coordinate \(x_{2}\). Use the fact that \(L\) and \(M\) are parallel to show that the second coordinate of \(Q\) is \(y_{2}+b\) (c) Compute the slope of \(L\) using \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\) Compute the slope of \(M\) using the points \(P\) and \(Q .\) Verify that the two slopes are the same.