Heldl is preparing for the national gymnastics competition. The table shows the number of hours Heldl spent preparing for a gymnastics competition over a period of five months:
| Month | 1 | 2 | 3 | 4 | 5 |
| :---- | :-: | :-: | :-: | :-: | :-: |
| Hours | 2 | 3.5 | 5 | 6.5 | 8 |
Did Heidi increase the number of hours of practice linearly or exponentially?
A. Linearly, because the table shows an equal increase in hours for an equal increase in months
B. Exponentially, because the table shows an equal increase in hours for an equal increase in months
C. Linearly, because the table shows that hours increase by an equal factor for an equal increase in months
Answer (1)
Calculate the difference in hours between consecutive months.
The difference between month 2 and month 1 is 3.5 − 2 = 1.5 .
The difference between month 3 and month 2 is 5 − 3.5 = 1.5 .
Since the difference is constant (1.5), the increase is linear. The answer is Linearly, because the table shows an equal increase in hours for an equal increase in months. $\boxed{Linearly, because the table shows an equal increase in hours for an equal increase in months}
Explanation
Understanding the Problem We are given a table showing the number of hours Heldl spent preparing for a gymnastics competition over five months. We need to determine if the increase in hours is linear or exponential. A linear increase means the difference between consecutive values is constant, while an exponential increase means the ratio between consecutive values is constant.
Checking for Linear Increase To determine if the increase is linear, we calculate the difference in hours between consecutive months: Month 2 - Month 1: 3.5 − 2 = 1.5 Month 3 - Month 2: 5 − 3.5 = 1.5 Month 4 - Month 3: 6.5 − 5 = 1.5 Month 5 - Month 4: 8 − 6.5 = 1.5 Since the difference between consecutive months is constant (1.5 hours), the increase is linear.
Checking for Exponential Increase (Not Necessary) To determine if the increase is exponential, we would check if the ratio between consecutive months is constant. However, since we already found that the increase is linear, we don't need to check for exponential increase.
Conclusion Since the difference in hours between consecutive months is constant (1.5 hours), the increase is linear. Therefore, the correct answer is: Linearly, because the table shows an equal increase in hours for an equal increase in months
Examples
Understanding linear growth is crucial in many real-world scenarios. For instance, if you're saving money each month with a fixed deposit, the increase in your savings follows a linear pattern. Similarly, if a plant grows at a constant rate each day, its height increases linearly. Recognizing linear patterns helps in predicting future outcomes based on consistent changes.
