Chapter 1

Find the equation of the line described, giving it in slope-intercept form if possible. Perpendicular to \(x=3,\) passing through \((1,2)\)

Solve each problem. Solar Heater Production A company produces 10 solar heaters for \(\$ 7500 .\) The cost to produce 20 heaters is \(\$ 13,900\) (a) Express the cost \(y\) as a linear function of the number of heaters, \(x\) (b) Determine analytically the cost to produce 25 heaters. (c) Support the result of part (b) graphically.

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x-4 ; x=-2$$

Set your viewing window to \([-10,10]\) by \([-10,10]\) and then set Xscl to 0 and \(Y\) scl to \(0 .\) What do you notice? Make a conjecture as to how to set the screen with no tick marks on the axes.

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$5[1-(3-x)]=3(5 x+2)-7$$

Find the equation of the line described, giving it in slope-intercept form if possible. Perpendicular to \(y=-1,\) passing through \((-4,5)\)

Solve each problem. Cricket Chirping At \(68^{\circ} \mathrm{F}\), a certain species of cricket chirps 112 times per minute. At \(46^{\circ} \mathrm{F}\), the same cricket chirps 24 times per minute. (a) Express the number of chirps, \(y,\) as a linear function of the Fahrenheit temperature. (b) If the temperature is \(60^{\circ} \mathrm{F}\), how many times will the cricket chirp per minute? (c) If you count the number of cricket chirps in one-half minute and hear 40 chirps, what is the temperature?

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=5 x+6 ; x=-5$$

Set your viewing window to \([-50,50]\) by \([-50,50]\) Xscl to \(1,\) and \(Y\) scl to \(1 .\) Describe the appearance of the axes compared with those seen in the standard window. Why do you think they appear this way? How can you change your scale settings so that this "problem" is alleviated?

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$7 x-3[5 x-(5+x)]=1-4 x$$

Find the equation of the line described, giving it in slope-intercept form if possible. Passing through \((-2,4)\) and perpendicular to the line passing through \(\left(-5, \frac{1}{2}\right)\) and \(\left(-3, \frac{2}{3}\right)\)

Solve each problem. Appraised Value of a Home In \(2005,\) a house was purchased for \(\$ 120,000 .\) In \(2015,\) it was appraised for \(\$ 146,000\) (a) If \(x=0\) represents 2005 and \(x=10\) represents 2015 express the appraised value of the house, \(y,\) as a linear function of the number of years, \(x,\) after 2005 (b) What was the house worth in the year \(2012 ?\) (c) What does the slope of the line represent?

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=2 x^{2}-x+3 ; x=1$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt{58}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$1.5(6 x-3)-7 x=3-(7-x)$$

Find the equation of the line described, giving it in slope-intercept form if possible. Passing through \(\left(\frac{3}{4}, \frac{1}{4}\right)\) and perpendicular to the line passing through \((-3,-5)\) and \((-4,0)\)

Solve each problem. Depreciation of a Photocopier \(\quad\) A photocopier sold for \(\$ 3000\) in 2008 . Its value in 2016 had depreciated to \(\$ 600\). (a) If \(x=0\) represents 2008 and \(x=8\) represents 2016 express the value of the machine, \(y,\) as a linear function of the number of years, \(x,\) after 2008 (b) Graph the function from part (a) in a window \([0,10]\) by \([0,4000] .\) How would you interpret the \(y\) -intercept in terms of this particular situation? (c) Use a calculator to determine the value of the machine in \(2012,\) and verify the result analytically.

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x^{2}+2 x-5 ; x=2$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt{97}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$-4[6-(-2+3 x)]=21+12 x$$

Find the equation of the line described, giving it in slope-intercept form if possible. Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-4,2)\) and \((2,10)\)

Solve each problem. Climate Change If the global climate were to warm significantly, the Arctic ice cap would start to melt. This ice cap contains an estimated \(680,000\) cubic miles of water. More than 200 million people currently live on land that is less than 3 feet above sea level. In the United States several large cities have low average elevations. Two examples are Boston (14 feet) and San Diego (13 feet). In this exercise you are to estimate the rise in sea level if this cap were to melt and determine whether this event would have a significant impact on people. (a) The surface area of a sphere is given by the expression \(4 \pi r^{2},\) where \(r\) is its radius. Although the shape of the earth is not exactly spherical, it has an average radius of 3960 miles. Estimate the surface area of the earth. (b) Oceans cover approximately \(71 \%\) of the total surface area of the earth. How many square miles of the earth's surface are covered by oceans? (c) Approximate the potential rise in sea level by dividing the total volume of the water from the ice cap by the surface area of the oceans. Convert your answer from miles to feet. (d) Discuss the implications of your calculation. How would cities such as Boston and San Diego be affected? (e) The Antarctic ice cap contains \(6,300,000\) cubic miles of water. Estimate how much sea level would rise if this ice cap melted. (Source: Department of the Interior, Geological Survey.)

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-x^{2}+x+2 ; x=4$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt[3]{33}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$-3[-5-(-9+2 x)]=2(3 x-1)$$

Find the equation of the line described, giving it in slope-intercept form if possible. Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-3,5)\) and \((4,9)\)

Solve each problem. Speeding Fines Suppose that speeding fines are determined by \(y=10(x-65)+50, x>65,\) where \(y\) is the cost in dollars of the fine if a person is caught driving \(x\) miles per hour. (a) How much is the fine for driving 76 mph? (b) While balancing the checkbook, Johnny found a check that his wife Gwen had written to the Department of Motor Vehicles for a speeding fine. The check was written for \(\$ 100 .\) How fast was Gwen driving? (c) At what whole-number speed are tickets first given? (d) For what speeds is the fine greater than \(\$ 200 ?\)

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-x^{2}-x-6 ; x=3$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt[3]{91}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$\frac{1}{2} x-2(x-1)=2-\frac{3}{2} x$$

Solve each problem. Expansion and Contraction of Gases \(\quad\) In \(1787,\) Jacques Charles noticed that gases expand when heated and contract when cooled. A particular gas follows the model2\(y=\frac{5}{3} x+455\) where \(x\) is the temperature in Celsius and \(y\) is the volume in cubic centimeters. (a) What is the volume when the temperature is \(27^{\circ} \mathrm{C} ?\) (b) What is the temperature when the volume is 605 cubic centimeters? (c) Determine what temperature gives a volume of 0 cubic centimeters.

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=5 ; x=9$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt[4]{86}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$0.5(x-2)+12=0.5 x+11$$

Match each equation with the graph in choices A-H that it most closely resembles. A.(GRAPHS CANNOT COPY) B.(GRAPHS CANNOT COPY) C.(GRAPHS CANNOT COPY) D.(GRAPHS CANNOT COPY) E.(GRAPHS CANNOT COPY) F.(GRAPHS CANNOT COPY) G.(GRAPHS CANNOT COPY) H.(GRAPHS CANNOT COPY) $$y=-3$$

Solve each problem. Sales of \(C R T\) and \(L C D\) Screens In the early 21 st century, LCD monitors were a new technology that replaced CRT (cathode ray tube) monitors. In \(2002,75\) million CRT monitors were sold and only 29 million flat LCD (liquid crystal display) monitors were sold. By 2006 the numbers were 45 million for CRT monitors and 88 million for LCD monitors. (a) Find a linear function \(C\) that models these data for CRT monitors and another linear function \(L\) that models these data for LCD monitors. Let \(x\) be the year. (b) Determine the year when sales of these two types of monitors were equal.

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-4 ; x=12$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt[4]{123}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$\frac{x-1}{2}=\frac{3 x-2}{6}$$

Investment problems such as those in Exercises \(61-66\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\). where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Real-Estate Financing A man wishes to sell a piece of property for \(\$ 240,000 .\) He wants the money to be paid off in two ways: a short-term note at \(6 \%\) interest and a longterm note at \(5 \%\). Find the amount of each note if the total annual interest paid is \(\$ 13,000\)

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt{x^{3}+12} ; x=-2$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$19^{1 / 2}$$

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$\frac{2 x-1}{3}=\frac{2 x+1}{3}$$

Investment problems such as those in Exercises \(61-66\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\). where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Buying and Selling Land A man bought two plots of land for a total of \(\$ 120,000 .\) When he sold the first plot, he made a profit of \(15 \% .\) When he sold the second, he lost 10\%. His total profit was \(\$ 5500 .\) How much did he pay for each piece of land?

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt[3]{x^{2}-x+6} ; x=2$$

Find a decimal approximation of each root or power. Round answers to the nearest thousandth. \(29^{1 / 3}\)

Global casino gaming revenue from online betting was \(\$ 147\) billion in 2012 and \(\$ 183\) billion in 2015 . (Source: Statista.) (a) Find an equation of a line \(y=m x+b\) that models this information, where \(y\) is in billions of dollars and \(x\) is the year. (b) Use this equation to estimate online betting revenue in 2014

Investment problems such as those in Exercises \(61-66\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\). where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Retirement Planning In planning her retirement, a woman deposits some money at \(2.5 \%\) interest with twice as much deposited at \(3 \% .\) Find the amount deposited at each rate if the total annual interest income is \(\$ 850\).

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt[3]{x^{2}}, x=8$$

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$46^{1.5}$$