The population P of a culture of bacteria is described by the equation P = 1600e^0.052t

where t is time, in hours, relative to the time at which the population was 1600.
(a) What was the population at t = 6 hours? Show your work.
(b) After how many hours will the population reach 8000? Round to the nearest tenth of an hour.
Show your work.

Please do not post any more questions until you've gone back shown what you know about solving your previous questions.

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(a) To find the population at t = 6 hours, we can substitute t = 6 into the equation P = 1600e^0.052t:

P = 1600e^0.052(6)

Calculating this, we get:

P = 1600e^0.312

Using a calculator, we find that e^0.312 is approximately 1.366. Therefore, we can find the population by multiplying 1600 by 1.366:

P ≈ 1600 * 1.366 ≈ 2185.6

So, the population at t = 6 hours is approximately 2185.6.

(b) To find the time it takes for the population to reach 8000, we can set P = 8000 in the equation P = 1600e^0.052t and solve for t:

8000 = 1600e^0.052t

Dividing both sides by 1600, we get:

5 = e^0.052t

To isolate e^0.052t, we can take the natural logarithm (ln) of both sides:

ln(5) = ln(e^0.052t)

Using the property ln(e^x) = x, we get:

ln(5) = 0.052t

Solving for t, we divide both sides by 0.052:

t = ln(5) / 0.052

Using a calculator, we find that ln(5) / 0.052 is approximately 26.7. Therefore, the population will reach 8000 after approximately 26.7 hours. Rounded to the nearest tenth of an hour, the population will reach 8000 after approximately 26.7 hours.

To find the population at t = 6 hours, we need to substitute t = 6 into the given equation.

(a) Population at t = 6 hours:
P = 1600e^(0.052t)
P = 1600e^(0.052 * 6)
P = 1600e^(0.312)

Using a calculator, we can evaluate e^(0.312) ≈ 1.365.

P = 1600 * 1.365
P ≈ 2184

Therefore, the population at t = 6 hours is approximately 2184.

To find the time it takes for the population to reach 8000, we can set P equal to 8000 and solve for t.

(b) Population at P = 8000:
P = 1600e^(0.052t)
8000 = 1600e^(0.052t)

Divide both sides by 1600:
5 = e^(0.052t)

Take the natural logarithm of both sides:
ln(5) = 0.052t

Divide by 0.052:
t = ln(5) / 0.052

Using a calculator, we can evaluate ln(5) ≈ 1.609.

t = 1.609 / 0.052
t ≈ 30.9

Rounding to the nearest tenth of an hour, after approximately 30.9 hours, the population will reach 8000.