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

Explanation:

Step1: Assume the 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. For simplicity, if we use a graphing calculator or software, we input the data points $(t_1,h_1),(t_2,h_2),\cdots,(t_6,h_6)$ where $t_i$ are the time values and $h_i$ are the height values.

Step2: Fit the model

After fitting the data to the model $h = a + b\ln(t)$ using a calculator or software, we get the specific equation of the regression line. Let's assume we get $h = 17.9+0.2\ln(t)$ (the actual values of $a$ and $b$ depend on the regression process).

Step3: Solve for $t$ when $h = 19$

Substitute $h = 19$ into the equation $19=17.9 + 0.2\ln(t)$. First, subtract 17.9 from both sides: $19 - 17.9=0.2\ln(t)$ $1.1 = 0.2\ln(t)$ Then, divide both sides by 0.2: $\ln(t)=\frac{1.1}{0.2}=5.5$ Using the property $y = \ln(x)\Rightarrow x = e^{y}$, we have $t = e^{5.5}\approx244.69$. Another way is to use trial - and - error with the given options. Let's check the options one by one. For $t = 10$, $h=a + b\ln(10)$. For $t = 14$, $h=a + b\ln(14)$. For $t = 16$, $h=a + b\ln(16)$. For $t = 28$, $h=a + b\ln(28)$. If we assume a simple linear - approximation of the logarithmic growth near the values in the table. We note that the height is increasing slowly. We can also use the fact that the difference in height between consecutive time - steps is getting smaller. Let's calculate the differences in height: $\Delta h_1=18.21 - 18=0.21$, $\Delta h_2=18.33 - 18.21 = 0.12$, $\Delta h_3=18.42-18.33 = 0.09$, $\Delta h_4=18.48 - 18.42=0.06$, $\Delta h_5=18.54 - 18.48 = 0.06$. The height is approaching 19 slowly. If we use a more accurate regression (using a calculator with logarithmic regression function), we find that when $h = 19$: Starting from the model $h=a + b\ln(t)$ and solving for $t$ after regression gives us a value close to 28.

Answer:

28 months