QuestionJune 6, 2025

A species with an initial population of 120 is growing in an environment where the carrying capacity is 750. After 3 years the population is up to 450 Use a logistic function to estimate the population after 5 years . Round to the nearest whole number. Population=[?] individuals.

A species with an initial population of 120 is growing in an environment where the carrying capacity is 750. After 3 years the population is up to 450 Use a logistic function to estimate the population after 5 years . Round to the nearest whole number. Population=[?] individuals.
A species with an initial population of 120 is growing in an environment where the carrying capacity is 750. After 3 years the population is up to 450 Use a logistic function to estimate the population after 5 years . Round to the nearest whole number. Population=[?] individuals.

Solution
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Answer

573 individuals. Explanation 1. Define the Logistic Growth Model The logistic growth model is given by P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}, where P(t) is the population at time t, K is the carrying capacity, P_0 is the initial population, and r is the growth rate. 2. Identify Known Values Initial population P_0 = 120, carrying capacity K = 750, population after 3 years P(3) = 450. 3. Solve for Growth Rate r Use P(3) = \frac{750}{1 + \left(\frac{750 - 120}{120}\right)e^{-3r}} = 450. Solve for r. Simplify to 450 = \frac{750}{1 + 5.25e^{-3r}}. Rearrange to find e^{-3r} = \frac{750}{450} - 1 = \frac{1}{3}. Thus, -3r = \ln\left(\frac{1}{3}\right), so r = -\frac{\ln\left(\frac{1}{3}\right)}{3}. 4. Calculate Population After 5 Years Use P(5) = \frac{750}{1 + \left(\frac{750 - 120}{120}\right)e^{-5r}}. Substitute r from Step 3 into the equation. Compute P(5) using the calculated r.

Explanation

1. Define the Logistic Growth Model<br /> The logistic growth model is given by $P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}$, where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $P_0$ is the initial population, and $r$ is the growth rate.<br /><br />2. Identify Known Values<br /> Initial population $P_0 = 120$, carrying capacity $K = 750$, population after 3 years $P(3) = 450$.<br /><br />3. Solve for Growth Rate $r$<br /> Use $P(3) = \frac{750}{1 + \left(\frac{750 - 120}{120}\right)e^{-3r}} = 450$. Solve for $r$.<br /> Simplify to $450 = \frac{750}{1 + 5.25e^{-3r}}$.<br /> Rearrange to find $e^{-3r} = \frac{750}{450} - 1 = \frac{1}{3}$.<br /> Thus, $-3r = \ln\left(\frac{1}{3}\right)$, so $r = -\frac{\ln\left(\frac{1}{3}\right)}{3}$.<br /><br />4. Calculate Population After 5 Years<br /> Use $P(5) = \frac{750}{1 + \left(\frac{750 - 120}{120}\right)e^{-5r}}$.<br /> Substitute $r$ from Step 3 into the equation.<br /> Compute $P(5)$ using the calculated $r$.
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