Question

example 3

1. measurement the table shows a function modeling the number of inches and feet. can the relationship be modeled by a linear or nonlinear function? explain.

feet	0	1	2	3	4
inches	0	12	24	36	48

1. astronomy the table shows the velocity of cassini 2 space probe as it passes saturn. is the velocity modeled by a linear or nonlinear function? explain.

time (s)	5	10	15	20	25
velocity (mph)	50,000	60,000	70,000	60,000	50,000

Explanation:

Step1: Recall the definition of linear function

A linear function has a constant rate of change. For a function $y = f(x)$, the rate of change is $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Analyze the relationship between feet and inches

Let $x$ be the number of feet and $y$ be the number of inches.
For the first two - points $(x_1 = 0,y_1 = 0)$ and $(x_2 = 1,y_2 = 12)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{12 - 0}{1 - 0}=12$.
For the points $(x_1 = 1,y_1 = 12)$ and $(x_2 = 2,y_2 = 24)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{24 - 12}{2 - 1}=12$.
For the points $(x_1 = 2,y_1 = 24)$ and $(x_2 = 3,y_2 = 36)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{36 - 24}{3 - 2}=12$.
For the points $(x_1 = 3,y_1 = 36)$ and $(x_2 = 4,y_2 = 48)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{48 - 36}{4 - 3}=12$.
Since the rate of change is constant ($12$ inches per foot), the relationship between feet and inches can be modeled by a linear function.

Step3: Analyze the velocity - time relationship of Cassini 2

Let $x$ be the time in seconds and $y$ be the velocity in mph.
For the first two - points $(x_1 = 5,y_1 = 50000)$ and $(x_2 = 10,y_2 = 60000)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{60000 - 50000}{10 - 5}=\frac{10000}{5}=2000$.
For the points $(x_1 = 10,y_1 = 60000)$ and $(x_2 = 15,y_2 = 70000)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{70000 - 60000}{15 - 10}=\frac{10000}{5}=2000$.
For the points $(x_1 = 15,y_1 = 70000)$ and $(x_2 = 20,y_2 = 60000)$, the rate of change is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{60000 - 70000}{20 - 15}=\frac{- 10000}{5}=-2000$.
Since the rate of change is not constant, the velocity of the Cassini 2 space - probe cannot be modeled by a linear function.

Answer:

1. The relationship between feet and inches can be modeled by a linear function because the rate of change (12 inches per foot) is constant.
2. The velocity of the Cassini 2 space - probe cannot be modeled by a linear function because the rate of change of velocity with respect to time is not constant.