the height of a plant over time is shown in the table below. using a logarithmic model, what is the best…

Explanation:

Step1: Assume logarithmic model

Let the logarithmic model be $h = a + b\ln(t)$. Using two - point method (for simplicity, taking the first two points $(t_1 = 1,h_1=18)$ and $(t_2 = 2,h_2 = 18.21)$). When $t = 1$, $h=18$, so $18=a + b\ln(1)$. Since $\ln(1)=0$, we get $a = 18$.

Step2: Find the value of b

Substitute $a = 18$, $t = 2$ and $h = 18.21$ into $h=a + b\ln(t)$. We have $18.21=18 + b\ln(2)$. Then $b=\frac{18.21 - 18}{\ln(2)}=\frac{0.21}{\ln(2)}\approx\frac{0.21}{0.693}\approx0.303$. So the model is $h = 18+0.303\ln(t)$.

Step3: Solve for t when h = 19

Set $h = 19$ in the model $19=18 + 0.303\ln(t)$. Then $19-18=0.303\ln(t)$, so $\ln(t)=\frac{1}{0.303}\approx3.3$. Taking the exponential of both sides, $t = e^{3.3}\approx27.11\approx28$ months.

Answer:

28 months