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Math  /  Calculus

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The rate of change dPdt\frac{d P}{d t} of the number of bacteria in a tank is modeled by a logistic differential equation. The maximum capacity of the tank is 914 bacteria. At 8 AM , the number of bacteria in the tank is 228 and is increasing at a rate of 37 bacteria per minute. Write a differential equation to describe the situation.
Answer Attempt 1 out of 2 dPdt=\frac{d P}{d t}= □\square Submit Answer

Studdy Solution

STEP 1

1. The rate of change of the number of bacteria is modeled by a logistic differential equation.
2. The maximum capacity of the tank is 914 bacteria.
3. At 8 AM, the number of bacteria is 228.
4. The rate of increase of bacteria at 8 AM is 37 bacteria per minute.

STEP 2

1. Recall the general form of the logistic differential equation.
2. Identify the parameters from the problem.
3. Write the specific logistic differential equation for this situation.

STEP 3

Recall the general form of the logistic differential equation:
dPdt=rP(1−PK) \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)
where: - P P is the population size (number of bacteria), - r r is the intrinsic growth rate, - K K is the carrying capacity (maximum capacity of the tank).

STEP 4

Identify the parameters from the problem: - The carrying capacity K K is given as 914 bacteria. - At 8 AM, P=228 P = 228 and dPdt=37 \frac{dP}{dt} = 37 .

STEP 5

Write the specific logistic differential equation for this situation:
Given that at P=228 P = 228 , dPdt=37 \frac{dP}{dt} = 37 , we can substitute these values into the logistic equation to find r r :
37=r×228(1−228914) 37 = r \times 228 \left(1 - \frac{228}{914}\right)
Solving for r r :
37=r×228×(1−228914) 37 = r \times 228 \times \left(1 - \frac{228}{914}\right)
Calculate the term (1−228914) \left(1 - \frac{228}{914}\right) :
1−228914=686914 1 - \frac{228}{914} = \frac{686}{914}
Substitute back:
37=r×228×686914 37 = r \times 228 \times \frac{686}{914}
r=37×914228×686 r = \frac{37 \times 914}{228 \times 686}
Calculate r r :
r≈0.237 r \approx 0.237
Now, write the differential equation:
dPdt=0.237P(1−P914) \frac{dP}{dt} = 0.237P\left(1 - \frac{P}{914}\right)
The differential equation describing the situation is:
dPdt=0.237P(1−P914) \boxed{\frac{dP}{dt} = 0.237P\left(1 - \frac{P}{914}\right)}

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