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? plant height t, time in months h, height in inches 1 18 2 18.21 3 18.33 4 18.42 5 18.48 6 18.54 10 months 14 months 16 months 28 months

Explanation:

Step1: Analyze the growth trend

The plant's height increases slowly over time. A logarithmic model \(h = a + b\ln(t)\) can be used.

Step2: Use approximation

We can observe that the height is increasing gradually. If we assume a simple logarithmic - like growth pattern (since we don't have the exact formula derivation from the table data for a full - fledged logarithmic regression).
We note that the height at \(t = 6\) is \(h=18.54\). The height \(h = 19\) is a small increase from \(h = 18.54\).
If we consider the rate of growth (the difference in height per month is getting smaller as \(t\) increases).
If we assume a very rough linear - like approximation of the logarithmic growth (for the sake of estimation from the given data points).
The increase from \(h = 18.54\) (at \(t = 6\)) to \(h = 19\) is \(\Delta h=19 - 18.54 = 0.46\).
Looking at the previous differences: \(18.21-18 = 0.21\), \(18.33 - 18.21=0.12\), \(18.42-18.33 = 0.09\), \(18.48 - 18.42=0.06\), \(18.54 - 18.48 = 0.06\).
If we assume that the growth rate continues to slow down.
If we consider that from \(t = 6\) to \(t = 16\) (a larger time interval), the cumulative growth can reach \(h = 19\).
Another way is to think about the fact that if we assume the model \(h(t)=18+\frac{\ln(t)}{k}\) (a very basic form of a logarithmic model).
When \(t = 1\), \(h(1) = 18\). When \(t = 2\), \(h(2)=18+\frac{\ln(2)}{k}=18.21\), then \(\frac{\ln(2)}{k}=0.21\), \(k=\frac{\ln(2)}{0.21}\approx3.3\).
If \(h(t)=19\), then \(19 = 18+\frac{\ln(t)}{3.3}\), \(\frac{\ln(t)}{3.3}=1\), \(\ln(t)=3.3\), \(t = e^{3.3}\approx27\) (but this is a very rough theoretical approach).
However, looking at the options and the slow - growth nature of the data (the height increases by \(0.54\) in \(6\) months from \(t = 1\) to \(t = 6\)).
If we assume that to get an increase of \(1\) (from \(h = 18\) to \(h = 19\)), and considering the decreasing rate of growth, \(t = 16\) is a more reasonable estimate among the given options than \(t = 10\) (too fast growth assumption) or \(t = 28\) (assuming too slow growth compared to the initial 6 - month trend).

Answer:

16 months