The rate of change (dP)/(dt) of the number of people on an island is modeled by a logistic differential equation. The maximum capacity of the island is 926 people. At 11AM, the number of people on the island is 135 and is increasing at a rate of 28 people per hour. Write a differential equation to describe the situation. (dP)/(dt)=◻" Submit Answer "

Question

The rate of change  (dP)/(dt) of the number of people on an island is modeled by a logistic differential equation. The maximum capacity of the island is 926 people. At  11AM, the number of people on the island is 135 and is increasing at a rate of 28 people per hour. Write a differential equation to describe the situation.  (dP)/(dt)=◻" Submit Answer "

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Logistic Differential Equation: The logistic differential equation is generally given by the formula: $$ \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) $$ where $ P $ is the population at time $ t $, $ r $ is the intrinsic growth rate, and $ K $ is the carrying capacity of the environment. In this case, $ K $ is given as 926 people.Given Population and Rate: We are given that at 11AM, the number of people on the island is 135, which is $ P $ at $ t = 11AM $, and the rate of increase $ \frac{dP}{dt} $ is 28 people per hour. However, we need to find the intrinsic growth rate $ r $ to complete the differential equation.Calculate Intrinsic Growth Rate: To find the intrinsic growth rate $ r $, we can use the given rate of increase when $ P = 135 $. Plugging these values into the logistic equation, we get: $$ 28 = r \cdot 135\left(1 - \frac{135}{926}\right) $$ Now we need to solve for $ r $.Calculate Fraction of Carrying Capacity: First, calculate the fraction of the carrying capacity: $$ 1 - \frac{135}{926} = 1 - 0.14579 \approx 0.85421 $$Calculate Term: Now, multiply this fraction by 135 to find the term $ 135\left(1 - \frac{135}{926}\right) $: $$ 135 \cdot 0.85421 \approx 115.3185 $$Calculate Intrinsic Growth Rate: Next, divide the rate of increase by this term to find $ r $: $$ r = \frac{28}{115.3185} \approx 0.2428 \text{ per hour} $$Complete Differential Equation: Now that we have $ r $, we can write the complete logistic differential equation: $$ \frac{dP}{dt} = 0.2428P\left(1 - \frac{P}{926}\right) $$ This is the differential equation that describes the situation.

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