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In a given culture, there are 1000 bacteria at any given time. After 10 minutes, there are 2000. How many bacteria will there be in 1 hour, knowing that they increase according to the formula P = P0 . ^ , where P is the number of bacteria, t is the time in hours, and k is a constant?

Answer 1

To find the number of bacteria after 1 hour, we need to use the formula P = P0 * e^(kt), where P is the final number of bacteria, P0 is the initial number of bacteria, t is the time in hours, and k is a constant.

Given that there are 1000 bacteria at the start (P0 = 1000) and after 10 minutes (t = 10/60 = 1/6 hours), the number of bacteria becomes 2000, we can substitute these values into the formula and solve for k.

2000 = 1000 * e^(k * 1/6)

Dividing both sides by 1000:

2 = e^(k/6)

To isolate e^(k/6), we take the natural logarithm (ln) of both sides:

ln(2) = k/6

Now, we can solve for k by multiplying both sides by 6:

k = 6 * ln(2)

Now that we have the value of k, we can use it to find the number of bacteria after 1 hour (t = 1):

P = P0 * e^(kt)

P = 1000 * e^(6 * ln(2) * 1)

P = 1000 * e^(6 * ln(2))

P ≈ 1000 * 2.718^(6 * 0.693)

P ≈ 1000 * 2.718^4.158

P ≈ 1000 * 64.498

P ≈ 64,498

Therefore, there will be approximately 64,498 bacteria after 1 hour.

Step-by-step explanation: