Calculus with Applications

News Sources A survey asked respondents where they got news "yesterday" (the day before they participated in the survey). In $2006,$ about 40$\%$ of respondents got their news from the newspaper, while 23$\%$ got their neir news online. In $2012,$ about 29$\%$ got their news from the newspaper, while 39$\%$ got their news online. Both news sources changed at a roughly linear rate.

Source: State of the Media.

(a) Find a linear equation expressing the percent of respon- dents, $y_{n},$ who got their news from the newspaper in terms of $t,$ the years since 2000 .

(b) Find a linear equation expressing the percent of respondents, $y_{o},$ who got their news online in terms of $t$ , the years since $2000 .$

(c) Find the rate of change over time for the percentage of

respondents for each source of news.

Calculus with Applications

Global Warming In $1990,$ the Intergovernmental Panel on Climate Change predicted that the average temperature on Earth would rise $0.3^{\circ} \mathrm{C}$ per decade in the absence of international controls on greenhouse emissions. Let $t$ measure the time in years since $1970,$ when the average global temperature was $15^{\circ} \mathrm{C} .$ Source: Science News.

(a) Find a linear equation giving the average global temperature in degrees Celsius in terms of $t$ , the number of years since 1970 .

(b) Scientists have estimated that the sea level will rise by 65 $\mathrm{cm}$ if the average global temperature rises to $19^{\circ} \mathrm{C}$ . According to your answer to part (a), when would this

occur?

Calculus with Applications

Galactic Distance The table lists the distances (in megaparsecs where 1 megaparsec $\approx 3.1 \times 10^{19} \mathrm{km}$ ) and velocities (in kilometers per second) of four galaxies moving rapidly away from Earth. Source: Astronomical Methods and Calculations, and Fundamental Astronomy.

$$\begin{array}{|c|c|c|}\hline \text { Galaxy } & {\text { Distance }} & {\text { Velocity }} \\ \hline \text { Virga } & {15} & {1600} \\ \hline \text { Ursa Minor } & {200} & {15,000} \\ \hline \\ {\text { Corona Borealis }} & {290} & {24,000} \\ \hline \\ {\text { Bootes }} & {520} & {40,000} \\ \hline \end{array}$$

(a) Plot the data points, letting $x$ represent distance and $y$ represent velocity. Do the points lie in an approximately linear pattern?

(b) Write a linear equation $y=m x$ to model these data, using the ordered pair $(520,40,000)$ .

(c) The galaxy Hydra has a velocity of $60,000 \mathrm{km}$ per sec. Use your equation to approximate how far away it is from Earth.

(d) The value of $m$ in the equation is called to estimate constant. The Hubble constant can be used to estimate the age of the universe $A$ (in years) using the formula

$$A=\frac{9.5 \times 10^{11}}{m}.$$

Approximate $A$ using your value of $m$.

Calculus with Applications

Marriage The following table lists the U.S. median age at first marriage for men and women. The age at which both groups marry for the first time seems to be increasing at a roughly linear rate in recent decades. Let $t$ correspond to the number of years since $1980 .$ Source: U.S. Census Bureau.

(a) Find a linear equation that approximates the data for men, using the data for the years 1980 and $2010 .$

(b) Repeat part (a) using the data for women.

(c) Which group seems to have the faster increase in median age at first marriage?

(d) According to the equation from part (a), in what year will the men's median age at first marriage reach 30$?$

(e) When the men's median age at first marriage is $30,$ what will the median age be for women?

Calculus with Applications

Immigration In 1950 , there were $249,187$ immigrants admitted to the United States. In $2012,$ the number was $1,031,631$ Source: 2012 Yearbook of Immigration Statistics.

(a) Assuming that the change in immigration is linear, write an equation expressing the number of immigrants, $y,$ in terms of $t$ , the number of years after 1900 .

(b) Use your result in part (a) to predict the number of immigrants admitted to the United States in $2015 .$

(c) Considering the value of the $y$ -intercept in your answer to part (a), discuss the validity of using this equation to model the number of immigrants throughout the entire 20 th century.

Calculus with Applications

Child Mortality Rate The mortality rate for children under 5 years of age around the world has been declining in a roughly linear fashion in recent years. The rate per 1000 live births was 90 in 1990 and 48 in $2012 .$ Source: World Health Organization.

(a) Determine a linear equation that approximates the mortality rate in terms of time $t,$ where $t$ represents the number of years since $1900 .$

(b) One of the Millennium Development Goals (MDG) of the World Health Organization is to reduce the mortality rate for children under 5 years of age to 30 by $2015 .$ If this trend were to continue, in what year would this goal be reached?

Calculus with Applications

Life Expectancy Some scientists believe there is a limit to how long humans can live. One supporting argument is that during the past century, life expectancy from age 65 has increased more slowly than life expectancy from birth, so eventually these two will be equal, at which point, according to these scientists, life expectancy should increase no further. In $1900,$ life expectancy at birth was 46 yr, and life expectancy at age 65 was 76 yr. In 2010 , these figures had risen to 78.7 and $84.1,$ respectively. In both cases, the increase in life expectancy has been linear. Using these assumptions and the data given, find the maximum life expectancy for humans. Source: Science.

Calculus with Applications

Ponies Trotting A study found that the peak vertical force on a trotting Shetland pony increased linearly with the pony's speed, and that when the force reached a critical level, the

pony switched from a trot to a gallop. For one pony, the critical force was 1.16 times its body weight. It experienced a force of 0.75 times its body weight at a speed of 2 meters per second and a force of 0.93 times its body weight at 3 meters per second. At what speed did the pony switch from a trot to a gallop? Source: Science.

Calculus with Applications

Exercise Heart Rate To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minute) should be in the target heart rate zone. The lower limit of this zone is found by taking 70$\%$ of the difference between 220 and your age. The upper limit is found by using 85$\% .$ Source: Physical Fitness.

(a) Find formulas for the upper and lower limits $(u$ and $l)$ as linear equations involving the age $x .$

(b) What is the target heart rate zone for a 20 -year-old?

(c) What is the target heart rate zone for a 40 -year-old?

(d) Two women in an aerobics class stop to take their pulse and are surprised to find that they have the same pulse. One woman is 36 years older than the other and is working at the upper limit of her target heart rate zone. The younger woman is working at the lower limit of her target heart rate zone. What are the ages of the two women, and what is their pulse?

(e) Run for 10 minutes, take your pulse, and see if it is in your target heart rate zone. (After all, this is listed as an exercise!)

Calculus with Applications

HIV Infection The time interval between a person's initial infection with HIV and that person's eventual development of AIDS symptoms is an important issue. The method of infection with HIV affects the time interval before AIDS develops. One study of HIV patients who were infected by intravenous drug use found that 17$\%$ of the patients had AIDS after 4 years, and 33$\%$ had developed the disease after 7 years. The relationship between the time interval and the percentage of patients with AlDS can be modeled accurately with a linear equation. Source: Epidemiologic Review.

(a) Write a linear equation $y=m t+b$ that models these data, using the ordered pairs $(4,0.17)$ and $(7,0.33) .$

(b) Use your equation from part (a) to predict the number of years before half of these patients will have AIDS.

Calculus with Applications

Tuition The table lists the annual cost (in dollars) of tuition and fees at private four-year colleges for selected years. (See Example $13 .$ Source: The College Board.

$$\begin{array}{ll}{\text { Year }} & {\text { Tuition and Fees }} \\ {2000} & {16,072} \\ {2002} & {18,060}\\{2004} & {20,045} \\ {2006} & {22,308} \\ {2008} & {24,818} \\ {2010} & {26,766} \\ {2012} & {28,989} \\ {2013} & {30,094}\end{array}$$

(a) Sketch a graph of the data. Do the data appear to lie roughly along a straight line?

(b) Let $t=0$ correspond to the year $2000 .$ Use the points $(0,16,072)$ and $(13,30,094)$ to determine a linear equation that models the data. What does the slope of the graph of the equation indicate?

(c) Discuss the accuracy of using this equation to estimate the cost of private college in $2025 .$

Calculus with Applications

cost The total cost for a bakery to produce 150 gourmet cupcakes is $\$ 225,$ while the total cost to produce 175 gourmet cupcakes is $\$ 247 .$ Let $y$ represent the total cost for $x$ gourmet cupcakes. Assume that a straight line can approximate the data

(a) Find and interpret the slope of the cost line for the gourmet cupcakes.

(b) Determine an equation that models the data. Write the equation in the slope-intercept form.

(c) Use your answer from part (b) to determine an approximation of the cost of 200 gourmet cupcakes.