Questions: With r=0.2, K=4 and Y0=3, use a calculator to verify that Y1=3.2 and then compute Y2, Y3, Y4, Y5, Y6, and Y7. Fill in the values in Table 1 which is given below, rounding to one decimal place.

With r=0.2, K=4 and Y0=3, use a calculator to verify that Y1=3.2 and then compute Y2, Y3, Y4, Y5, Y6, and Y7. Fill in the values in Table 1 which is given below, rounding to one decimal place.

Transcript text: With $r=0.2, K=4$ and $Y_{0}=3$, use a calculator to verify that $Y_{1}=3.2$ and then compute $Y_{2}, Y_{3}, Y_{4}, Y_{5}, Y_{6}$, and $Y_{7}$. Fill in the values in Table 1 which is given below, rounding to one decimal place.

Solution

Solution Steps

To solve this problem, we need to use the logistic growth model formula to compute the yeast population for each subsequent year. The logistic growth model is given by:

\[ Y_{n+1} = Y_n + r \cdot Y_n \cdot \left(1 - \frac{Y_n}{K}\right) \]

where:

$ Y_n $ is the population at year $ n $
$ r $ is the intrinsic growth rate
$ K $ is the carrying capacity

Given $ r = 0.2 $, $ K = 4 $, and $ Y_0 = 3 $, we can compute $ Y_1 $ through $ Y_7 $ using the formula iteratively.

Step 1: Initial Values and Formula

Given:

$ r = 0.2 $
$ K = 4 $
$ Y_0 = 3 $

The logistic growth model formula is: \[ Y_{n+1} = Y_n + r \cdot Y_n \cdot \left(1 - \frac{Y_n}{K}\right) \]

Step 2: Compute $ Y_1 $ to $ Y_7 $

Using the formula iteratively, we compute the following values:

\[ \begin{align_} Y_1 &= 3 + 0.2 \cdot 3 \cdot \left(1 - \frac{3}{4}\right) = 3 + 0.2 \cdot 3 \cdot 0.25 = 3 + 0.15 = 3.2 \\ Y_2 &= 3.2 + 0.2 \cdot 3.2 \cdot \left(1 - \frac{3.2}{4}\right) = 3.2 + 0.2 \cdot 3.2 \cdot 0.2 = 3.2 + 0.128 = 3.3 \\ Y_3 &= 3.3 + 0.2 \cdot 3.3 \cdot \left(1 - \frac{3.3}{4}\right) = 3.3 + 0.2 \cdot 3.3 \cdot 0.175 = 3.3 + 0.1155 = 3.4 \\ Y_4 &= 3.4 + 0.2 \cdot 3.4 \cdot \left(1 - \frac{3.4}{4}\right) = 3.4 + 0.2 \cdot 3.4 \cdot 0.15 = 3.4 + 0.102 = 3.5 \\ Y_5 &= 3.5 + 0.2 \cdot 3.5 \cdot \left(1 - \frac{3.5}{4}\right) = 3.5 + 0.2 \cdot 3.5 \cdot 0.125 = 3.5 + 0.0875 = 3.6 \\ Y_6 &= 3.6 + 0.2 \cdot 3.6 \cdot \left(1 - \frac{3.6}{4}\right) = 3.6 + 0.2 \cdot 3.6 \cdot 0.1 = 3.6 + 0.072 = 3.7 \\ Y_7 &= 3.7 + 0.2 \cdot 3.7 \cdot \left(1 - \frac{3.7}{4}\right) = 3.7 + 0.2 \cdot 3.7 \cdot 0.075 = 3.7 + 0.0555 = 3.8 \\ \end{align_} \]

Final Answer

\[ \begin{array}{r|llllllll} \hline r & Y_0 & Y_1 & Y_2 & Y_3 & Y_4 & Y_5 & Y_6 & Y_7 \\ \hline 0.2 & 3 & 3.2 & 3.3 & 3.4 & 3.5 & 3.6 & 3.7 & 3.8 \\ \hline \end{array} \]

$\boxed{Y_0 = 3, Y_1 = 3.2, Y_2 = 3.3, Y_3 = 3.4, Y_4 = 3.5, Y_5 = 3.6, Y_6 = 3.7, Y_7 = 3.8}$

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