Question

Tree Diameter For a certain type of tree the diameter D (in feet) depends on the tree’s age t (in years) according to the logistic growth model

$D\left(t\right)=\frac{5.4}{1+2.9e^{-0.01t}}$

Find the diameter of a 20-year-old tree.

Short answer

The diameter of the 20-year-old tree is 1.38 feet.

Step-by-step solution

Step 1. Given.

For a certain type of tree, the diameter D (in feet) depends on the tree’s age t (in years) according to the logistic growth model

$D\left(t\right)=\frac{5.4}{1+2.9e^{-0.01t}}$

Step 2. To determine.

We have to find; find the diameter of a 20-year-old tree.

Step 3. Calculation.

According to the question, $t=20$.

We plug$t=20$ in the given function:

$$\begin{gathered} D\left(t\right)=\frac{5.4}{1+2.9e^{-0.01t}} \\ D\left(0\right)=\frac{5.4}{1+2.9e^{-0.01\left(0\right)}} \\ =\frac{5.4}{1+2.9e^0} \\ =\frac{5.4}{1+2.9\left(1\right)} \\ =\frac{5.4}{1+2.9} \\ =\frac{5.4}{3.9} \\ =1.38461538462 \\ \approx 1.38 \end{gathered}$$

Hence, the diameter of the 20-year-old tree is 1.38 feet.