To determine whether a linear or exponential function is more appropriate for modeling the height of the tennis ball after several bounces, we need to analyze the pattern in the given data.
The data provided is:
- Bounce number [tex]\( n \)[/tex]: 0, 1, 2, 3, 4
- Height [tex]\( h \)[/tex] (in cm): 150, 79.5, 42.2, 22.4, 11.9
### Step 1: Check for a Linear Pattern
A linear function has the form: [tex]\( h = mn + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
To check if the data follows a linear pattern, we need to see if the differences between consecutive heights are approximately constant.
Calculating the differences:
- [tex]\( h_1 - h_0 = 79.5 - 150 = -70.5 \)[/tex]
- [tex]\( h_2 - h_1 = 42.2 - 79.5 = -37.3 \)[/tex]
- [tex]\( h_3 - h_2 = 22.4 - 42.2 = -19.8 \)[/tex]
- [tex]\( h_4 - h_3 = 11.9 - 22.4 = -10.5 \)[/tex]
These differences [tex]\( (-70.5, -37.3, -19.8, -10.5) \)[/tex] are not constant, indicating the heights do not follow a linear pattern.
### Step 2: Check for an Exponential Pattern
An exponential function has the form: [tex]\( h = h_0 \cdot r^n \)[/tex], where [tex]\( h_0 \)[/tex] is the initial height and [tex]\( r \)[/tex] is the common ratio.
To verify if the data follows an exponential pattern, we need to see if the ratios of consecutive heights are approximately constant.
Calculating the ratios:
- [tex]\( \frac{h_1}{h_0} = \frac{79.5}{150} \approx 0.53 \)[/tex]
- [tex]\( \frac{h_2}{h_1} = \frac{42.2}{79.5} \approx 0.53 \)[/tex]
- [tex]\( \frac{h_3}{h_2} = \frac{22.4}{42.2} \approx 0.53 \)[/tex]
- [tex]\( \frac{h_4}{h_3} = \frac{11.9}{22.4} \approx 0.53 \)[/tex]
These ratios [tex]\( (0.53, 0.53, 0.53, 0.53) \)[/tex] are approximately constant, indicating the heights follow an exponential pattern.
### Conclusion:
Since the ratios of consecutive heights are approximately constant, the most appropriate model for the height [tex]\( h \)[/tex] of the tennis ball after [tex]\( n \)[/tex] bounces is an exponential function. Thus, the correct choice is:
Exponential is more appropriate because there is a common factor.
