Chapter 1

Average Wages The average hourly wage (adjusted to 1982 dollars) was \(\$ 7.66\) in 1990 and \(\$ 9.16\) in 2017. (a) Find a point-slope form of a line that passes through the points \((1990,7.66)\) and \((2017,9.16)\) (b) Interpret the slope. (c) Use the equation from part (a) to approximate the hourly wage in \(2009 .\) Compare it with the actual value of \(\$ 8.27\)

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. $$23^{2.75}$$

The table shows equivalent temperatures in degrees Celsius and degrees Fahrenheit. $$\begin{array}{|c|c|c|c|c|c|} \hline^{\circ} \mathbf{F} & -40 & 32 & 59 & 95 & 212 \\ \hline^{\circ} \mathrm{C} & -40 & 0 & 15 & 35 & 100 \end{array}$$ (a) Plot the data by having the \(x\) -axis correspond to Fahrenheit temperature and the \(y\) -axis to Celsius temperature. What type of relation exists between the data? (b) Find a function \(C\) that uses the Fahrenheit temperature \(x\) to calculate the corresponding Celsius temperature. Interpret the slope. (c) Convert a temperature of \(86^{\circ} \mathrm{F}\) to degrees Celsius.

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. Lottery Winnings A woman won \(\$ 200,000\) in a state lottery. She first paid income tax of \(30 \%\) on the winnings. Of the rest, she invested some at \(1.5 \%\) and some at \(4 \%\) earning \(\$ 4350\) interest per year. How much did she invest at each rate?

For the given \(f(x)\) find (a) \(f(a),\) (b) \(f(b+1),\) and (c) \(f(3 x)\). $$f(x)=5 x$$

Approximate each expression to the nearest hundredth. $$\frac{5.6-3.1}{8.9+1.3}$$

Asian-American populations in the United States (in millions) are shown in the table. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2007 & 2009 & 2011 & 2013 \\ \hline \begin{array}{l} \text { Population } \\ \text { (in millions) } \end{array} & 13.2 & 13.3 & 14.3 & 16.4 \end{array}$$ (a) Use the points \((2007,13.2)\) and \((2013,16.4)\) to find the point-slope form of a line that models the data. Let \(\left(x_{1}, y_{1}\right)\) be \((2007,13.2)\) (b) Use this equation to estimate the Asian-American population in 2014 to the nearest tenth of a million.

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. Cookbook Royalties \(A\) woman earned \(\$ 48,000\) from royalties on her cookbook. She paid a \(28 \%\) income tax on these royalties. The balance was invested in two ways, some of it at \(3.25 \%\) interest and some at \(1.75 \% .\) The investments produced \(\$ 904.80\) interest the first year. Find the amount invested at each rate.

Approximate each expression to the nearest hundredth. $$\frac{34+25}{23}$$

The table lists the worldwide advertising revenue of Google (in billions of dollars). $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2012 & 2014 & 2016 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { ( 5 billions) } \end{array} & 28 & 44 & 60 & 79 \\ \hline \end{array}$$ (a) Find the point-slope form of the line that passes through the points \((2010,28)\) and \((2016,79) .\) Let \(\left(x_{1}, y_{1}\right)\) be \((2010,28)\) (b) Interpret the slope of the line. (c) Use this equation to estimate the revenue in 2012 and \(2014 .\) Compare these estimates with the actual values shown in the table.

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

Approximate each expression to the nearest hundredth. $$\sqrt{\pi^{3}+1}$$

For the given \(f(x)\) find (a) \(f(a),\) (b) \(f(b+1),\) and (c) \(f(3 x)\). $$f(x)=2 x-5$$

The table lists U.S. print newspaper advertising revenue (in billions of dollars). $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2009 & 2011 & 2013 & 2014 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { ( 5 billions) } \end{array} & 25 & 21 & 17 & 16 \\ \hline \end{array}$$ (a) Find the point-slope form of the line that passes though \((2009,25)\) and \((2014,16) .\) Let \(\left(x_{1}, y_{1}\right)\) be \((2009,25)\) (b) Interpret the slope of the line. (c) Use this equation to estimate the revenue for 2012

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

Approximate each expression to the nearest hundredth. $$\sqrt[3]{2.1-6^{2}}$$

The table lists the average total tuition, fees, room, and board costs (in constant 2014 dollars) at private colleges and universities for selected years. $$\begin{array}{|l|c|c|c|c} \hline \text { Year } & 1984 & 1994 & 2004 & 2014 \\ \hline \begin{array}{l} \text { Tuition and Fees } \\ \text { (in 2014 dollars) } \end{array} & 18,354 & 25,502 & 31,875 & 37,424 \end{array}$$ (a) Find the equation of the least-squares regression line that models the data. (b) Graph the data and the regression line in the same viewing window. (c) Estimate tuition and fees in \(2005,\) and compare it with the actual value of \(\$ 29,307\)

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

Approximate each expression to the nearest hundredth. $$3(5.9)^{2}-2(5.9)+6$$

The table lists the average tuition and fees (in constant 2014 dollars) at public colleges and universities for selected years. $$\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1984 & 1994 & 2004 & 2014 \\ \hline \begin{array}{l} \text { Tuition and Fees } \\ \text { (in 2014 dollars) } \end{array} & 7626 & 9386 & 12,179 & 16,188 \\ \hline \end{array}$$ (a) Find the equation of the least-squares regression line that models the data. (b) Graph the data and the regression line in the same viewing window. (c) Estimate the cost in 2009 (d) Use the model to predict the cost in 2019 .

Solve each formula for the specified variable. \(P=a+b+c\) for \(c \quad\) (Perimeter of a triangle)

Approximate each expression to the nearest hundredth. $$2 \pi^{3}-5 \pi^{2}-3$$

In the late 1920 s, the famous observational astronomer Edwin \(P\). Hubble \((1889-1953)\) determined the distances to several galaxies and the velocities at which they were receding from Earth. Four galaxies with their distances in light-years and velocities in miles per second are listed in the table below. $$\begin{array}{|l|c|c|} \hline \multicolumn{1}{|c|}\text { Galaxy } & \text { Distance } & \text { Velocity } \\ \hline \text { Virgo } & 50 & 990 \\ \text { Ursa Minor } & 650 & 9,300 \\ \text { Corona Borealis } & 950 & 15,000 \\ \text { Bootes } & 1,700 & 25,000 \end{array}$$ (a) Let \(x\) represent velocity and \(y\) represent distance. Find the equation of the least-squares regression line that models the data. (b) If the galaxy Hydra is receding at a speed of \(37,000\) miles per second, estimate its distance from Earth.

Solve each formula for the specified variable. \(s l=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) for \(h \quad\) (Area of a trapezoid)

Approximate each expression to the nearest hundredth. $$\sqrt{(4-6)^{2}+(7+1)^{2}}$$

Work each problem. If \(f(-2)=3,\) identify a point on the graph of \(f\)

The table lists the worldwide average household spending (in dollars) on Apple products for selected years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2009 & 2011 & 2013 & 2015 \\ \hline \begin{array}{l} \text { Spending } \\ \text { (in dollars) } \end{array} & 62 & 158 & 265 & 444 \\ \hline \end{array}$$ (a) Use regression to find a formula \(f(x)=a x+b\) so that \(f\) models the data. (b) Interpret the slope of the graph of \(y=f(x)\) (c) Estimate the average household spending on Apple products in 2014 and compare it with the actual value of \(\$ 343\)

Solve each formula for the specified variable. \(d=\frac{1}{2} h\left(b_{1}+b_{2}\right)\) for \(b_{2} \quad\) (Area of a trapezoid)

Approximate each expression to the nearest hundredth. $$\sqrt{[-1-(-3)]^{2}+(-5-3)^{2}}$$

Work each problem. If \(f(3)=-9.7,\) identify a point on the graph of \(f\)

Obtain the least squares regression line and the correlation coefficient. Make \(a\) statement about the correlation. Gestation Period and Life Span of Animals $$\begin{array}{|l|c|c|} \text { Animal } & \begin{array}{c} \text { Average Gestation } \\ \text { or Incubation Period, } \\ \boldsymbol{x} \text { (in days) } \end{array} & \begin{array}{c} \text { Record Life } \\ \text { Span, } \boldsymbol{y} \\ \text { (in years) } \end{array} \\ \hline \text { Cat } & 63 & 26 \\ \text { Dog } & 63 & 24 \\ \text { Duck } & 28 & 15 \\ \text { Elephant } & 624 & 71 \\ \text { Goat } & 151 & 17 \\ \text { Guinea pig } & 68 & 6 \\ \text { Hippopotamus } & 240 & 49 \\ \text { Horse } & 336 & 50 \\ \text { Lion } & 108 & 29 \\ \text { Parakeet } & 18 & 12 \\ \text { Pig } & 115 & 22 \\ \text { Rabbit } & 31 & 15 \\ \text { Sheep } & 151 & 16 \end{array}$$

Solve each formula for the specified variable. \(S=2 L W+2 W H+2 H L\) for \(H \quad\) (Surface area of a rectangular box )

Approximate each expression to the nearest hundredth. $$\frac{\sqrt{\pi-1}}{\sqrt{1+\pi}}$$

Work each problem. If the point \((7,8)\) lies on the graph of \(f,\) then \(F\) (_____) = (____).

Obtain the least squares regression line and the correlation coefficient. Make \(a\) statement about the correlation. Urban Areas in the World (Population and Area) $$\begin{array}{|l|c|c|} \text { Urban Area } &\begin{array}{c} \text { Population, } \\ x \text { (in millions) } \end{array} & \begin{array}{c} \text { Area, } y \\ \text { (in square } \\ \text { miles) } \end{array} \\ \hline \text { Tokyo-Yokohama, Japan } & 34.2 & 3287 \\ \text { Delhi, India } & 23.9 & 747 \\ \text { Mumbai, India } & 23.3 & 210 \\ \text { Mexico City, Mexico } & 22.8 & 787 \\ \text { New York, United States } & 22.2 & 4477 \\ \text { São Paulo, Brazil } & 20.8 & 1220 \\ \text { Shanghai, China } & 18.8 & 1345 \\ \text { Los Angeles, United States } & 17.9 & 2423 \\ \text { Kolkata (Calcutta), India } & 16.6 & 463 \\ \text { Buenos Aires, Argentina } & 13.1 & 1016 \end{array}$$

Solve each formula for the specified variable. \(S=2 \pi r h+2 \pi r^{2}\) for \(h \quad\) (Surface area of a cylinder)

Approximate each expression to the nearest hundredth. $$\sqrt[3]{4.5 \times 10^{5}+3.7 \times 10^{2}}$$

\(f\) and \(g\) are linear functions. If the solution set of \(f(x) \geq g(x)\) is \([4, \infty),\) what is the solution set of each equation or inequality? (a) \(f(x)=g(x)\) (b) \(f(x)>g(x) \quad\) (c) \(f(x)

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(-1,3), m=\frac{3}{2}$$

Solve each formula for the specified variable. \(V=\frac{1}{3} \pi r^{2} h\) for \(h \quad\) (Volume of a cone)

Approximate each expression to the nearest hundredth. $$\frac{2}{1-\sqrt[3]{5}}$$

\(f\) and \(g\) are linear functions. If the solution set of \(f(x)

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(-2,8), m=-1$$

Solve each formula for the specified variable. \(y=a(x-h)^{2}+k\) for \(a \quad\) (Mathematics)

Approximate each expression to the nearest hundredth. $$1-\frac{4.5}{3-\sqrt{2}}$$

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(3 x-6=0\) (b) \(3 x-6>0\) (c) \(3 x-6<0\)

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(3,-4), m=-\frac{1}{3}$$

Solve each formula for the specified variable. \(S=\frac{n}{2}\left(a_{1}+a_{n}\right)\) for \(n \quad\) (Mathematics)

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(5 x+10=0\) (b) \(5 x+10>0\) (c) \(5 x+10<0\)