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
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(1228914) 37 = r \times 228 \left(1 - \frac{228}{914}\right)
Solving for r r :
37=r×228×(1228914) 37 = r \times 228 \times \left(1 - \frac{228}{914}\right)
Calculate the term (1228914) \left(1 - \frac{228}{914}\right) :
1228914=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 :
r0.237 r \approx 0.237
Now, write the differential equation:
dPdt=0.237P(1P914) \frac{dP}{dt} = 0.237P\left(1 - \frac{P}{914}\right)
The differential equation describing the situation is:
dPdt=0.237P(1P914) \boxed{\frac{dP}{dt} = 0.237P\left(1 - \frac{P}{914}\right)}

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