Question

the height of a plant over time is shown in the table below. using a logarithmic model, what is the best estimate for the age of the plant when it is 19 inches tall?

t, time in months	plant height
2	18
3	18.21
4	18.33
5	18.42
6	18.48
	18.54

10 months
14 months
16 months
28 months

Explanation:

Step1: Assume a logarithmic model

Let the logarithmic model be $h = a + b\ln(t)$. We can use the method of least - squares or a graphing utility with regression capabilities. Here, we can also make a rough estimate by observing the trend. The height is increasing slowly as time goes on.

Step2: Analyze the growth rate

The height is increasing by small amounts each month. From $t = 2$ to $t=6$, the height increases from $18$ to $18.54$. The difference between $19$ and $18$ is $1$. Since the growth is slow, we can expect it to take a relatively long time to reach $19$ inches.

Step3: Estimate the time

By looking at the slow - growth pattern, we note that the growth rate is getting smaller as $t$ increases. If we consider the options, $10$ months is too soon as the growth in the first few months is not enough to reach $19$ inches. Among the options, $28$ months seems to be the most reasonable estimate as the plant is growing slowly and steadily.

Answer:

28 months