Eleen has a greenhouse and is growing orchids. The table shows the average number of orchids that bloomed over a period of four months:

\begin{tabular}{|l|l|l|l|l|}
\hline Month & 1 & 2 & 3 & 4 \\
\hline Orchids & 11 & 12.4 & 13.8 & 15.2 \\
\hline
\end{tabular}

Did the number of orchids increase linearly or exponentially?

A. Linearly, because the table shows that the orchids increased by the same amount each month.

B. Linearly, because the table shows a constant percentage increase in orchids each month.

C. Exponentially, because the table shows that the orchids increased by the same amount each month.

D. Exponentially, because the table shows a constant percentage increase in orchids each month.



Answer :

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.