- Use the model \(h(t) = a + b \ln(t)\) with data point \((1, 18)\) to find \(a = 18\).
- Use the point \((6, 18.54)\) to find \(b = \frac{0.54}{\ln(6)} \approx 0.3014\).
- Set \(h(t) = 19\) and solve for \(t\) using \(1 = b \ln(t)\) to find \(t \approx 27.61\) months.
- The closest provided option is \(28\) months.
- Final answer: \(\boxed{28}\).
Explanation
- Analyzing the Logarithmic Model
The plant height \(h\) in inches is modeled as a function of time \(t\) in months using the logarithmic equation \(h(t) = a + b \ln(t)\). We are given data points \((t, h)\): \((1, 18), (2, 18.21), (3, 18.33), (4, 18.42), (5, 18.48), (6, 18.54)\). Our goal is to find the age \(t\) when the plant height \(h(t)\) reaches \(19\) inches.
- Determining the Constants a and b
First, we determine the constant \(a\) using the data point for \(t = 1\):
\[h(1) = a + b \ln(1) = 18.\]
Since \(\ln(1) = 0\), we have \(a + b(0) = 18\), which simplifies to \(a = 18\). Next, we find the constant \(b\) using the data point for \(t = 6\):
\[h(6) = 18 + b \ln(6) = 18.54.\]
Subtracting \(18\) from both sides gives \(b \ln(6) = 0.54\), so \(b = \frac{0.54}{\ln(6)}\). Using the natural logarithm of \(6 \approx 1.79176\), we find \(b \approx \frac{0.54}{1.79176} \approx 0.3014\).
- Solving for the Age t and Concluding
Now we set the height \(h(t) = 19\) to solve for \(t\):
\[19 = 18 + b \ln(t).\]
Subtracting \(18\) from both sides, we get:
\[1 = b \ln(t).\]
Substituting \(b = \frac{0.54}{\ln(6)}\) into the equation:
\[1 = \frac{0.54}{\ln(6)} \ln(t).\]
Rearranging to solve for \(\ln(t)\):
\[\ln(t) = \frac{\ln(6)}{0.54} \approx \frac{1.79176}{0.54} \approx 3.318.\]
Taking the exponential of both sides gives \(t = e^{3.318} \approx 27.61\). Comparing this result to the given options (10, 14, 16, or 28 months), the value \(27.61\) is closest to \(28\).
Examples
Logarithmic models are frequently used in biological growth studies. For example, when observing the growth of a population of bacteria or the height of trees over time, the rate of growth often slows down as the organism matures, a pattern well-captured by logarithmic curves. By using such models, researchers can project future growth or estimate the time it took to reach a certain size, which is critical in fields like forestry, environmental science, and medical research.