Chapter 3

Solve each problem. The rising base price \(P\) (in dollars) for a new Ford \(\mathrm{F} 150\) can be modeled by the function \(P=793 n+15,950,\) where \(n\) is the number of years since 2000. a) What will be the base price for a new Ford F150 in \(2009 ?\) b) By what amount is the price increasing annually? c) Graph the equation for \(0 \leq n \leq 10\)

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has \(y\) -intercept \((0,5)\) and \(x\) -intercept \((4,0)\).

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-3,-1)\) and is parallel to \(y=2 x+6\).

Solve each problem. For a one-day car rental the X-press Car Company charges \(C\) dollars, where \(C\) is determined by the function \(C=0.26 m+42\) and \(m\) is the number of miles driven. a) What is the charge for a car driven 400 miles? b) Sketch a graph of the equation for \(m\) ranging from 0 to 1000

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((1,-3)\) and is parallel to \(y=-3 x-5\).

Solve each problem. The Friendly Bob Loan Company gives each applicant a rating, \(t,\) from 0 to 10 according to the applicant's ability to repay, a higher rating indicating higher risk. The interest rate, \(r\), is then determined by the function \(r=0.02 t+0.15\) a) If your rating were \(8,\) then what would be your interest rate? b) Sketch the graph of the equation for \(t\) ranging from 0 to 10

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) is parallel to \(2 x+4 y=1\) and goes through \((-3,5)\).

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(0) $$

Solve each problem. The function \(C=0.50 t+8.95\) gives the customer's cost in dollars for a pan pizza, where \(t\) is the number of toppings. a) Find the cost of a five-topping pizza. b) Find \(t\) if \(C=14.45\) and interpret your result.

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) is parallel to \(3 x-5 y=-7\) and goes through \((-8,-2)\).

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-1,-2)\) and is perpendicular to \(y=-3 x+7\).

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(4) $$

Solve each problem. An office manager is placing an order for note pads at \(\$ 1\) each and binders at \(\$ 2\) each. The total cost of the order must be \(\$ 100 .\) Write an equation for the total cost and graph it. If he orders 30 note pads, then how many binders must he order?

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-1,-2)\) and is perpendicular to \(y=-3 x+7\).

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(100) $$

Solve each problem. Jessenda is ordering tacos at \(\$ 0.75\) each and burritos at \(\$ 2\) each for a large group. She must spend \(\$ 300 .\) Write an equation for the total cost and graph it. If she orders 200 tacos, then how many burritos must she order?

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-4,-3)\) and is perpendicular to \(x+3 y=4\).

Solve each problem. Hillary sells roses at a busy Los Angeles intersection. The functions$$\begin{array}{l}C=0.55 x+50 \\\R=1.50 x\end{array}$$and$$P=0.95 x-50$$ give her weekly cost, revenue, and profit in terms of \(x\) where \(x\) is the number of roses that she sells in one week. a) Find \(C, R,\) and \(P\) if \(x=850 .\) Interpret your results. b) Find \(x\) if \(P=995\) and interpret your result. c) Find \(R-C\) if \(x=1100\) and interpret your result.

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) is perpendicular to \(2 y+5-3 x=0\) and goes through \((2,7)\).

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((2,5)\) and is parallel to the \(x\) -axis.

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ h(-3) $$

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-1,6)\) and is parallel to the \(y\) -axis.

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y=3 x-8, x+3 y=7$$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y=\frac{1}{2} x-4, \frac{1}{2} x+\frac{1}{4} y=1$$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 x-4 y=9, \frac{1}{3} x=\frac{2}{3} y-8$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(2)+g(3) $$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$\frac{1}{4} x-\frac{1}{6} y=\frac{1}{3}, \frac{1}{3} y=\frac{1}{2} x-2$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ f(1)-g(0) $$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 y=x+6, y-2 x=4$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ \frac{g(2)}{h(-3)} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$y-3 x=5,3 x+y=7$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ \frac{h(-10)}{f(2)} $$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$f(-1) \cdot h(-4)$$

Solve each problem. See Example \(8 .\) Waist-to-hip ratio. A study by Dr. Aaron R. Folsom concluded that waist-to-hip ratios are a better predictor of 5-year survival than more traditional height- to-weight ratios. Dr. Folsom concluded that for good health the waist size of a woman aged 50 to 69 should be less than or equal to \(80 \%\) of her hip size, \(w \leq 0.80 h .\) Make a graph showing possible waist and hip sizes for good health for women in this age group for which hip size is no more than 50 inches.

Determine whether each pair of lines is parallel, perpendicular, or neither. $$9-x=3, \frac{1}{2} x=8$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$h(0) \cdot g(0)$$

Solve each problem. See Example 9. Heating water. The temperature of a cup of water is a linear function of the time that it is in the microwave. The temperature at 0 seconds is \(60^{\circ} \mathrm{F}\) and the temperature at 120 seconds is \(200^{\circ} \mathrm{F}\) a) Express the linear function in the form \(t=m s+b\) where \(t\) is the Fahrenheit temperature and \(s\) is the time in seconds. [Hint: Write the equation of the line through \((0,60) \text { and }(120,200) .]\) b) Use the linear function to determine the temperature at 30 seconds. c) Graph the linear function.

Explain the difference between a compound inequality using the word and and a compound inequality using the word or.

Solve each problem. See Example 9. Carbon dioxide emission. Worldwide emission of carbon dioxide (CO \(_{2}\) ) increased linearly from 14 billion tons in 1970 to 26 billion tons in 2000 (World Resources Institute, www.wri.org). a) Express the emission as a linear function of the year in the form \(y=m x+b,\) where \(y\) is in billions of tons and \(x\) is the year. [ Hint: Write the equation of the line through \((1970,14) \text { and }(2000,26) .]\) b) Use the function from part (a) to predict the worldwide emission of \(\mathrm{CO}_{2}\) in 2010 .

Find a formula that expresses the area of a square \(A\) as a function of the length of its side \(s\).

Discussion Explain how to write an absolute value inequality as a compound inequality.

Find a formula that expresses the perimeter of a square \(P\) as a function of the length of its side \(s\).

Solve each problem. See Example 9. Depth and flow. When the depth of the water in the Tangipahoa River at Robert, Louisiana, is 9.14 feet, the flow is 1230 cubic feet per second ( \(\mathrm{ft}^{3} / \mathrm{sec}\) ). When the depth is 7.84 feet, the flow is \(826 \mathrm{ft}^{3} / \mathrm{sec} .\) (U.S. Geological Survey, www.usgs.gov). Let \(w\) represent the flow in cubic feet per second and \(d\) represent the depth in feet. a) Write the equation of the line through \((9.14,1230)\) and \((7.84,826)\) and express \(w\) in terms of \(d .\) Round to two decimal places. b) What is the flow when the depth is \(8.25 \mathrm{ft} ?\) c) Is the flow increasing or decreasing as the depth increases?

If a certain fabric is priced at \(\$ 3.98\) per yard, express the cost \(C(x)\) as a function of the number of yards \(x .\) Find \(C(3)\).

If Mildred earns \(\$ 14.50\) per hour, express her total pay \(P(h)\) as a function of the number of hours worked \(h .\) Find \(P(40)\).

Exploration The intercept form for the equation of a line is 2 \(\frac{x}{a}+\frac{y}{b}=1\) where neither \(a\) nor \(b\) is zero. a) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{4}+\frac{y}{6}=1\) b) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{a}+\frac{y}{b}=1\) c) Write the equation of the line through \((0,3)\) and \((-5,0)\) in intercept form. d) Which lines cannot be written in intercept form?

A pizza parlor charges \(\$ 14.95\) for a pizza plus \(\$ 0.50\) for each topping. Express the total cost of a pizza \(C(n)\) in dollars as a function of the number of toppings \(n .\) Find \(C(6)\)

A gravel dealer charges \(\$ 50\) plus \(\$ 30\) per cubic yard for delivering a truckload of gravel. Express the total cost \(C(n)\) in dollars as a function of the number of cubic yards delivered \(n .\) Find \(C(12)\).

Consider \(y=x+2\) and \(y>x+2 .\) Explain why one of these relations is a function and the other is not.

Graph \(y=2 x-400\) and \(y=-0.5 x+1\) on the same screen, using the viewing window \(-500 \leq x \leq 500\) and \(-1000 \leq y \leq 1000 .\) Should these lines be perpendicular? Explain.