To determine whether the number of orchids increased linearly or exponentially, we need to look at both the absolute increase in the number of orchids and the percentage increase each month.
### Calculating the Absolute Increase
First, we calculate the difference between consecutive months. This will help us see if the number of orchids shows a constant increase:
1. From month 1 to month 2: [tex]\( 12.4 - 11 = 1.4 \)[/tex]
2. From month 2 to month 3: [tex]\( 13.8 - 12.4 = 1.4 \)[/tex]
3. From month 3 to month 4: [tex]\( 15.2 - 13.8 = 1.4 \)[/tex]
The differences between consecutive months are [tex]\( 1.4, 1.4, \)[/tex] and [tex]\( 1.4 \)[/tex]. Since the increase in the number of orchids is the same each month (1.4), this suggests the increase is linear.
### Calculating the Percentage Increase
Next, we calculate the percentage increase from one month to the next:
1. From month 1 to month 2: [tex]\( \frac{12.4}{11} - 1 \approx 0.127 \times 100 \approx 12.73\% \)[/tex]
2. From month 2 to month 3: [tex]\( \frac{13.8}{12.4} - 1 \approx 0.113 \times 100 \approx 11.29\% \)[/tex]
3. From month 3 to month 4: [tex]\( \frac{15.2}{13.8} - 1 \approx 0.101 \times 100 \approx 10.14\% \)[/tex]
The percentage increases between consecutive months are approximately [tex]\( 12.73\%, 11.29\%, \)[/tex] and [tex]\( 10.14\% \)[/tex]. These values are not constant; they are decreasing over time.
### Conclusion
- Linearly: The absolute increase is constant because the number of orchids increased by the same amount (1.4) each month.
- Exponentially: The percentage increase is not constant as it varies (12.73%, 11.29%, 10.14%).
Therefore, the number of orchids did not increase exponentially.
The correct answer is:
- Linearly, because the table shows that the orchids increased by the same amount each month.
