Chapter 1

(Modeling) In Exercises \(53-56\), assume that a linear relationship exists between the two quantities. 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.

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. $$7 x-3[5 x-(5+x)]=1-4 x$$

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

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

(Modeling) In Exercises \(53-56\), assume that a linear relationship exists between the two quantities. 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?

Set your viewing window to \([-50,50]\) by \([-50,50]\) Xscl to \(1,\) and Yscl 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. $$5[1-(3-x)]=3(5 x+2)-7$$

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

Match each equation with the graph that it most closely resembles. $$y=-3 x-6$$

Passing through \((-2,4)\) and perpendicular to the line passing through \(\left(-5, \frac{1}{2}\right)\) and \(\left(-3, \frac{2}{3}\right)\).

(Modeling) In Exercises \(53-56\), assume that a linear relationship exists between the two quantities. Appraised Value of a Home In \(2002,\) a house was purchased for \(\$ 120,000 .\) In \(2012,\) it was appraised for \(\$ 146,000\) (a) If \(x=0\) represents 2002 and \(x=10\) represents 2012 express the appraised value of the house, \(y,\) as a linear function of the number of years, \(x,\) after 2002 (b) What was the house worth in the year \(2009 ?\) hat does the slope of the line represent?

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. $$0.2(5 x-4)-0.1(6-3 x)=0.4$$

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

Passing through \(\left(\frac{3}{4}, \frac{1}{4}\right)\) and perpendicular to the line passing through \((-3,-5)\) and \((-4,0)\)

(Modeling) In Exercises \(53-56\), assume that a linear relationship exists between the two quantities. Depreciation of a Photocopier \(\quad\) A photocopier sold for \(\$ 3000\) in \(2006 .\) Its value in 2014 had depreciated to \(\$ 600 .\) (a) If \(x=0\) represents 2006 and \(x=8\) represents 2014 express the value of the machine, \(y,\) as a linear function of the number of years, \(x,\) after 2006 (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 your calculator to determine the value of the machine in 2010 , and verify your result analytically.

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. $$1.5(6 x-3)-7 x=3-(7-x)$$

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

Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-4,2)\) and \((2,10)\)

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 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. $$-4[6-(-2+3 x)]=21+12 x$$

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

Find the equation of the line that is the perpendicular bisector of the line segment connecting \((-3,5)\) and \((4,9)\)

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 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. $$-3[-5-(-9+2 x)]=2(3 x-1)$$

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

Expansion and Contraction of Gases In \(1787,\) Jacques Charles noticed that gases expand when heated and contract when cooled. A particular gas follows the model $$ y=\frac{5}{3} x+455 $$ 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 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. $$\frac{1}{2} x-2(x-1)=2-\frac{3}{2} x$$

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

Sales of CRT and LCD Screens In the early 21 st cenJury, LCD monitors were a new technology that replaced -CRT (cathode ray tube) monitors. In \(2002,75\) million CRT -nonitors were sold and only 29 million flat LCD (liquid erystal display) monitors were sold. By \(2006,\) the numbers -vere 45 million for CRT monitors and 88 million for LCD nonitors. (Source: International Data Corporation.) -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 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. $$0.5(x-2)+12=0.5 x+11$$

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

Solve each formula for the specified variable.} \(I=P R T\) for \(P \quad\) (Simple interest)

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. $$\frac{x-1}{2}=\frac{3 x-2}{6}$$

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

Solve each formula for the specified variable.} \(V=L W H\) for \(L \quad\) (Volume of a box)

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

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. $$\frac{2 x-1}{3}=\frac{2 x+1}{3}$$

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=|5-2 x|, x=8$$

Worldwide gambling revenue from online betting was \(\$ 18\) billion in 2007 and \(\$ 24\) billion in \(2010 .\) (Source: Christiansen Capital Advisors.) (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 2013.

Solve each formula for the specified variable.} \(P=2 L+2 W\) for \(W\) (Perimeter of a rectangle)

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

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\left|6-\frac{1}{2} x\right|, x=20$$