## Section1.1Variables

### SubsectionWhat is a Variable?

A variable is a numerical quantity that changes over time or in different situations.

We can show the values of a variable in a table or a graph.

###### Example1.1.

Life expectancy is the average age to which people will live. The table below shows that your life expectancy depends on the year of your birth. Thus, life expectancy is a variable.

 Year Born Life Expectancy $1900$ $49$ $1910$ $51$ $1920$ $58$ $1930$ $59$ $1940$ $63$ $1950$ $68$ $1960$ $70$ $1970$ $71$ $1980$ $74$ $1990$ $75$
1. To what age could people born in 1940 expect to live?

2. In what year did people's life expectancy reach 70 years of age?

Solution
1. People born in 1940 lived to 63 years of age on average.

2. Life expectancy reached 70 years of age in 1960.

###### Example1.2.

The graph below shows the average annual salaries of NFL football players, starting in 1940. The average salary of NFL football players is a variable. 1. Use the graph to estimate the average annual salary of NFL football players in 1975.

2. By how much did salaries increase from 1975 to 1980?

Solution
1. In 1975, the average NFL player's salary was $40,000. 2. In 1980, the average salary was$80,000, so salaries had increased from $40,000 to$80,000, or by

### ExercisesHomework 1.1

For Problems 1–4,

1. Find the pattern and fill in the table.

2. Write an equation for the second variable in terms of the first variable.

###### 1.
 $m$ $g$ $2$ $5$ $3$ $6$ $5$ $8$ $10$ $13$ $12$ $16$ $18$ $m$
###### 2.
 $t$ $w$ $0$ $20$ $2$ $18$ $4$ $16$ $5$ $15$ $6$ $10$ $12$ $t$
###### 3.
 $b$ $x$ $0$ $0$ $2$ $1$ $4$ $2$ $5$ $2.5$ $6$ $8$ $9$ $b$
###### 4.
 $z$ $3$ $6$ $8$ $12$ $15$ $18$ $20$ $z$ $r$ $2$ $4$ $\dfrac{16}{3}$ $8$

Hint: What fraction can you multiply by $z$ to get $r\text{?}$

For Problems 5–6, make your own table. Choose values for the first variable, and use the rule to find the values of the second variable.

###### 5.

$W = 1.2 \times n$

 $n$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $W$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$
###### 6.

$M = \dfrac{3}{2} \times x$

 $x$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $M$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$

For Problems 7–8, fill in the tables. What do you notice? Explain.

###### 7.
1. $y = \dfrac{x}{8}$
 $x$ $y$ $4$  $8$  $10$  $16$ 
2. $y = \dfrac{1}{8} x$

 $x$ $y$ $4$  $8$  $10$  $16$ 
3. $y = 0.125 \times x$

 $x$ $y$ $4$  $8$  $10$  $16$ 
###### 8.
1. $y = \dfrac{3 \times x}{5}$

 $x$ $y$ $5$  $10$  $12$  $1$ 
2. $y = \dfrac{3}{5} \times x$

 $x$ $y$ $5$  $10$  $12$  $1$ 
3. $y = 0.6 \times x$

 $x$ $y$ $5$  $10$  $12$  $1$ 
###### 9.

The temperature in Sunnyvale is usually $15 \degree$ hotter than it is in Ridgecrest, which is at a higher elevation. Fill in the table.

 Temperature in Ridgecrest $70$ $75$ $82$ $86$ $90$ $R$ Calculation Temperature in Sunnyvale
1. Explain in words how to find the temperature in Sunnyvale.
2. Write your explanation as a mathematical sentence:

$\text{Temp in Sunnyvale} =$

3. Let $R$ stand for the temperature in Ridgecrest and $S$ for the temperature in Sunnyvale. Write an equation for $S$ in terms of $R\text{.}$
4. Plot the points from the table and connect them with a smooth curve. ###### 10.

Jerome is driving from his home in White Falls to his parents' home in Castle Heights, 200 miles away. Fill in the table.

 Miles driven $40$ $60$ $90$ $120$ $170$ $d$ Calculation Miles remaining
1. Explain in words how to find the number of miles Jerome has left to drive.
2. Write your explanation as a mathematical sentence:

$\text{Miles remaining} =$

3. Let $d$ stand for the number of miles Jerome has driven and $r$ for the number of miles that remain. Write an equation for $r$ in terms of $d\text{.}$
4. Plot the points from the table and connect them with a smooth curve. ###### 11.

Milton goes to a restaurant with two friends, and they agree to split the bill equally. Fill in the table.

 Total bill $15$ $30$ $45$ $75$ $81$ $b$ Calculation Milton's share
1. Explain in words how to find Milton's share of the bill.
2. Write your explanation as a mathematical sentence:

$\text{Milton's share} =$

3. Let $b$ stand for the bill and $s$ for Milton's share. Write an equation for $s$ in terms of $b\text{.}$
4. Plot the points from the table and connect them with a smooth curve. ###### 12.

Nutrition experts tell us that no more than 30% of the calories we consume should come from fat. Fill in the table with the fat calories allowed daily for various calorie levels.

 Total calories $1000$ $1500$ $2000$ $2500$ $3000$ $C$ Calculation Fat calories
1. Explain in words how to find the number of fat calories allowed.
2. Write your explanation as a mathematical sentence:

$\text{Fat calories} =$

3. Let $C$ stand for the number of calories and $F$ for the number of fat calories. Write an equation for $F$ in terms of $C\text{.}$
4. Plot the points from the table and connect them with a smooth curve. Use the graphs to answer the questions in Problems 13–14. You may have to estimate some of your answers.

###### 13.

Suppose you invest $2000 in a retirement account that pays 8% interest compounded continuously. The amount of money in the account each year is shown by the graph below. 1. What variable is displayed on the horizontal axis? The vertical axis? 2. How much money will be in the account 5 years from now? 3. When will the amount of money in the account exceed$6000?
4. How much will the account grow between the second and third years?
5. How much will the account grow between the twelfth and thirteenth years?
###### 14.

Wendy brings a Thermos of soup for her lunch on a hike in the mountains. When she pours the soup into a bowl, it begins to cool off. The graph below shows the temperature of the soup after it is served. 1. What variable is displayed on the horizontal axis? The vertical axis?
2. How long does it take the soup to cool below 100?
3. What is the temperature of the soup after 8 minutes?
4. How many degrees does the soup cool in the first 2 minutes?
5. After a while, the temperature of the soup levels off at the same as the outside temperature. What was the outside temperature that day?
###### 15.

A cantaloupe is dropped from a tall building. Its speed increases until it hits the ground. Which of the four graphs below best represents the speed of the cantaloupe? Explain why your choice is the best one. ###### 16.
1. Match each of the following stories with the appropriate graph.

1. Delbert walks directly from home to school, then stays there.
2. Delbert walks towards school, but stops at a coffee shop for a cappuccino, then he goes on to school and stays there.
3. Delbert walks toward school, but decides to return home, where he stays.
2. Write your own story for the remaining graph. ###### 17.

Plot each pair of values on the grid provided. Connect the points with a smooth curve. Which graph appears to be a straight line?

1.  $t$ $0$ $1$ $2$ $5$ $6$ $8$ $v$ $16$ $14$ $12$ $6$ $4$ $0$ 2.  $w$ $0$ $1$ $2$ $4$ $6$ $10$ $Q$ $12$ $8$ $6$ $4$ $3$ $2$ ###### 18.
1. Read the graph at right to fill in the table.

 $x$ $0$ $10$ $30$ $40$ $60$ $70$ $y$
2. Find a formula expressing $y$ in terms of $x\text{.}$ For Problems 19–20,

1. Write an equation for the second variable in terms of the first variable.

2. Fill in the table. Plot the points and connect them with a smooth curve.
###### 19.

 $x$ $0$ $0.2$ $0.3$ $0.5$ $0.6$ $0.7$ $y$ $2$ $1.8$ $1.7$ ###### 20.
 $x$ $1$ $2$ $3$ $4$ $6$ $8$ $y$ $120$ $60$ $40$ 