Chapter 2

Modeling Sunrise Times In Boston, on the 90 th day (March 30 ) of 2008 the sun rose at 6: 30 A.M., and on the 129th day (May 8) the sun rose at 5:30 A.M. Use a linear function to estimate the days when the sun rose between 5: 45 A.M. and 6: 00 A.M. Do not consider days after May 8 .

Median Age The median age \(A\) in the United States during year \(x,\) where \(2000 \leq x \leq 2050,\) is projected to be $$ A(x)=0.07(x-2000)+35.3 $$ Use \(A(x)\) to estimate when the median age may reach 37 years. (Source: Bureau of the Census.)

Graph \(y=1000 x+1000\) in the standard window. (A) Is the graph a vertical line? (B) Explain why the calculator screen appears as it does.

Modelling Sunrise Times In Denver, on the 77 th day (March 17 ) of 2008 the sun rose at 7: 00 A.M., and on the 112 th day (April 21 ) the sun rose at 6: 00 A.M. Use a linear function to estimate the days when the sun rose between 6: 10 A.M. and 6: 40 A.M. Do not consider days after April 21.

Population Density In 1980 the population density of the United States was 64 people per square mile, and in 2000 it was 80 people per square mile. Use a linear function to estimate when the U.S. population density reached 87 people per square mile.

Graph the lines \(y=2 x\) and\(y=-\frac{1}{2} x\) in the standard viewing rectangle. (a) Do the lines appear to be perpendicular? (b) Graph the lines in the following viewing rectangles.i. \([-15,15,1]\) by \([-10,10,1]\) \ddoti. \([-10,10,1]\) by \([-3,3,1]\) in. \([-3,3,1]\) by \([-2,2,1]\) Do the lines appear to be perpendicular in any of these viewing rectangles? (c) Determine the viewing rectangles where perpendicular lines will appear perpendicular. (Answers may vary.)

Error Tolerances Suppose that an aluminum can is manufactured so that its radius \(r\) can vary from 1.99 inches to 2.01 inches. What range of values is possible for the circumference \(C\) of the can? Express your answer by using a three-part inequality.

Value of a Home In 1999 the value of a house was \(\$ 180,000,\) and in 2009 it was \(\$ 245,000\) (a) Find a linear function \(V\) that approximates the value of the house during year \(x .\) (b) What does the slope of the graph of \(V\) represent? (c) Use \(V\) to estimate the year when the house was worth \(\$ 219,000\)

A student graphs \(f(x)=x^{2}-x\) in the viewing rectangle \([2,2.1,0.01]\) by \([1.9,2.3,0.1] .\) Using the graph, the student decides that \(f\) is a linear function. How could you convince the student otherwise?

Error Tolerances Suppose that a square picture frame has sides that vary between 9.9 inches and 10.1 inches. What range of values is possible for the perimeter \(P\) of the picture frame? Express your answer by using a three-part inequality.

Sale Price \(\mathbf{A}\) store is discounting all regularly priced merchandise by \(25 \% .\) Find a function \(f\) that computes the sale price of an item having a regular price of \(x .\) If an item normally costs \(\$ 56.24\), what is its sale price?

Explain how average rate of change relates to a linear function.

The following data are exactly linear. $$ \begin{array}{cccc} x & 0 & 2 & 4 & 6 \\ y & -1.5 & 4.5 & 10.5 & 16.5 \end{array} $$ (a) Find a linear function \(f\) that models the data. (b) Solve the inequality \(f(x)>2.25\)

A rectangle is determined by the stated conditions. Find the slope-intercept form of the four lines that outline the rectangle. $$ \text { Vertices }(1,1),(5,1), \text { and }(5,5) $$

Find a real data set on the Internet that can be modeled by a linear function. Find the linear modeling function. Is your model exact or approximate? Explain.

The following data are exactly linear. $$ \begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 0.4 & 3.5 & 6.6 & 9.7 & 12.8 \end{array} $$ (a) Find a linear function \(f\) that models the data. (b) Solve the inequality \(2 \leq f(x) \leq 8\)

Skin Cancer Approximately \(4.5 \%\) of all cancer cases diagnosed in 2007 were skin cancer. (Source: American Cancer Society.) (a) If \(x\) cases of cancer were diagnosed, how many of these were skin cancer? (b) There were \(65,000\) cases of skin cancer diagnosed in \(2007 .\) Find the total number of cancer cases in 2007

A rectangle is determined by the stated conditions. Find the slope-intercept form of the four lines that outline the rectangle. $$ \text { Vertices }(4,0),(0,4),(0,-4), \text { and }(-4,0) $$

Explain how you determine whether a linear function is increasing, decreasing, or constant. Give an example of each.

Grades In order to receive an \(\mathrm{A}\) in a college course it is necessary to obtain an average of \(90 \%\) correct on three 1-hour exams of 100 points each and on one final exam of 200 points. If a student scores \(82,88,\) and 91 on the 1-hour exams, what is the minimum score that the person can receive on the final exam and still earn an A?

Explain what a piece wise-defined function is and why it is used. Sketch a graph of a continuous piecewiselinear function \(f\) that increases, decreases, and is constant. Let the domain of \(f\) be \(-4 \leq x \leq 4\)

Home Ownership Rates The table lists the percentage \(P\) of U.S. homes that are owned by their occupant rather than rented for selected years \(x .\) $$ \begin{array}{lllll} x & 1900 & 1950 & 1980 & 2006 \\ \hline P & 47 \% & 55 \% & 64 \% & 69 \% \end{array} $$ (a) Find a linear function \(P\) that models the data. (b) Estimate the years when this percentage was from \(58 \%\) to \(60 \%\) (c) Did your estimate involve interpolation or extrapolation?

Working Together Suppose that a lawn can be raked by one gardener in 3 hours and by a second gardener in 5 hours. (a) Mentally estimate how long it will take the two gardeners to rake the lawn working together. (b) Solve part (a) symbolically.

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=5,\) if \(y=7\) when \(x=14\)

Pumping Water Suppose that a large pump can empty a swimming pool in 50 hours and a small pump can empty the pool in 80 hours. How long will it take to empty the pool if both pumps are used?

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=2.5,\) if \(y=13\) when \(x=10\)

Motion A car went 372 miles in 6 hours, traveling part of the time at 55 miles per hour and part of the time at 70 miles per hour. How long did the car travel at each speed?

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=\frac{1}{2},\) if \(y=\frac{3}{2}\) when \(x=\frac{2}{3}\)

Mixing Candy Two types of candy sell for \(\$ 2.50\) per pound and \(\$ 4.00\) per pound. A store clerk is trying to make a 5 -pound mixture worth \(\$ 17.60 .\) How much of each type of candy should be included in the mixture?

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=1.3,\) if \(y=7.2\) when \(x=5.2\)

Running At 2: 00 PM. a runner heads north on a highway, jogging at 10 miles per hour. At 2: 30 PM. a driver heads north on the same highway to pick up the runner. If the car travels at 55 miles per hour, how long will it take the driver to catch the runner?

Investments \(\quad\) A total of \(\$ 5000\) was invested in two accounts. One pays \(5 \%\) annual interest, and the second pays \(7 \%\) annual interest. If the first-year interest is \(\$ 325\), how much was invested in each account?

Shadow Length \(\quad\) A person 66 inches tall is standing 15 feet from a streetlight. If the person casts a shadow 84 inches long, how tall is the streetlight?

Height of a Tree In the accompanying figure, a person 5 feet tall casts a shadow 4 feet long. A nearby tree casts a shadow 33 feet long. Find the height of the tree by solving a linear equation.(IMAGE CAN'T COPY)

Conical Water Tank A water tank in the shape of an inverted cone has a height of 11 feet and a radius of 3.5 feet, as illustrated in the figure. If the volume of the cone is \(V=\frac{1}{3} \pi r^{2} h,\) find the volume of the water in the tank when the water is 7 feet deep. (Hint: Consider using similar triangles.)(IMAGE CAN'T COPY)

The cost of tuition is directly proportional to the number of credits taken. If 11 credits cost \(\$ 720.50,\) find the cost of taking 16 credits. What is the constant of proportionality?

Dimension of a Cone (Refer to Exercise \(109 .\) ) A conical water tank holds 100 cubic feet of water and has a diameter of 6 feet. Estimate its height to the nearest tenth of a foot.

The maximum load that a horizontal beam can carry is directly proportional to its width. If a beam 1.5 inches wide can support a load of 250 pounds, find the load that a beam of the same type can support if its width is 3.5 inches.

Chemistry Determine how much pure water should be mixed with 5 liters of a \(40 \%\) solution of sulfuric acid to make a \(15 \%\) solution of sulfuric acid.

Stratospheric ozone occurs in the atmosphere between altitudes of 12 and 18 miles. Ozone in the stratosphere is frequently measured in Dobson units, where 300 Dobson units corresponds to an ozone layer 3 millimeters thick. In 1991 the reported minimum in the Antarctic ozone hole was about 110 Dobson units. (Source: R. Huffman, Armospheric Ultraviolet Remote Sensing. (a) The thickness \(y\) of the ozone layer is directly proportional to the number of Dobson units \(x\). Find the constant of proportionality \(k\) (b) How thick was the ozone layer in \(1991 ?\)

Mixing Antifreeze \(\quad\) A radiator holds 5 gallons of fluid. If it is full with a \(15 \%\) solution, how much fluid should be drained and replaced with a \(65 \%\) antifreeze mixture to result in a \(40 \%\) antifreeze mixture?

The weight of an object on Earth is directly proportional to the weight of an object on Mars. If a 25 -pound object on Earth weighs 10 pounds on Mars, how much would a 195 -pound astronaut weigh on Mars?

Window Dimensions A rectangular window has a length that is 18 inches more than its width. If its perimeter is 180 inches, find its dimensions.

Suppose a 15 -pound weight stretches a spring 8 inches, as shown in the figure. (A) Find the spring constant. (B) How far will a 25 -pound weight stretch this spring?

Online Holiday Shopping In 2003 online holiday sales were \(\$ 17\) billion, and in 2006 they were \(\$ 26\) billion. (Source: Digital Lifestyles.) (a) Find a linear function \(S\) that models these data. (b) Interpret the slope of the graph of \(S\). (c) Predict when online holiday sales might reach \(\$ 41\) billion.

If an 80 -pound force compresses a spring 3 inches, how much force must be applied to compress the spring 7 inches?

Sales of CRT and LCD Screens In \(2002,75\) million CRT (cathode ray tabe) monitors were sold and 29 million flat LCD (liquid crystal display) monitors were sold. In 2006 the numbers were 45 million for CRT monitors and 88 million for LCD monitors. (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) Interpret the slopes of the graphs of \(C\) and of \(L\) (c) Determine graphically the year when sales of these two types of monitors were equal. (d) Solve part (c) symbolically. (e) Solve part (c) numerically.

Geometry A 174 -foot-long fence is being placed around the perimeter of a rectangular swimming pool that has a 3 -foot-wide sidewalk around it. The actual swimming pool without the sidewalk is twice as long as it is wide. Find the dimensions of the pool without the sidewalk.

The electrical resistance of a wire varies directly with its length. If a 255 -foot wire has a resistance of 1.2 ohms, find the resistance of 135 feet of the same type of wire. Interpret the constant of proportionality in this situation.

Temperature Scales The Celsius and Fahrenheit scales are related by the equation \(C=\frac{5}{9}(F-32)\) These scales have the same temperature reading at a unique value where \(F=C .\) Find this temperature.