The population of a culture of bacteria, P(t), where t is time in days, is growing at a rate that is proportional to the population itself and the growth rate is 0.3. The initial population is 40.

(1) What is the population after 30 days?

(2) How long does it take for the population to double? (Round your answer to one decimal place.)
days

(1) After 30 days, the population can be calculated using the formula for exponential growth:

P(t) = P(0) * e^(rt)

where P(0) is the initial population, r is the growth rate, and e is Euler's number. In this case, P(0) = 40 and r = 0.3.

P(30) = 40 * e^(0.3 * 30)

Calculating this, we get:

P(30) ≈ 936.48

So, the population after 30 days is approximately 936.

(2) To find the time it takes for the population to double, we can use the formula:

P(t) = P(0) * e^(rt)

Since we know that the population doubles, we can set P(t) = 2P(0) and solve for t. In this case, P(0) = 40 and r = 0.3.

2P(0) = P(0) * e^(0.3t)

Now, we can divide both sides by P(0) and take the natural logarithm to solve for t:

ln(2) = 0.3t

t = ln(2) / 0.3

Calculating this, we get:

t ≈ 2.317

So, it takes approximately 2.3 days for the population to double.

To solve this problem, we can use the formula for exponential growth:

P(t) = P₀ * e^(r * t)

Where:
- P(t) is the population at time t
- P₀ is the initial population
- r is the growth rate
- t is the time in days

Given that the growth rate is 0.3 and the initial population is 40, we can plug these values into the formula.

(1) To find the population after 30 days:

P(30) = 40 * e^(0.3 * 30)

Calculating P(30):

P(30) = 40 * e^(9)

Using a scientific calculator, we find that e^9 ≈ 8103.08

So, P(30) = 40 * 8103.08 ≈ 324,123.2

Therefore, the population after 30 days is approximately 324,123.2.

(2) To find the time it takes for the population to double:

We know that when the population doubles, it will be equal to 2 * P₀.

Let's use the formula with P(t) = 2 * P₀ and solve for t:

2 * P₀ = P₀ * e^(r * t)

Divide both sides by P₀:

2 = e^(r * t)

Take the natural logarithm of both sides to isolate t:

ln(2) = (r * t)

Solve for t:

t = ln(2) / r

Calculating t:

t = ln(2) / 0.3 ≈ 2.3103

Therefore, it takes approximately 2.3 days for the population to double.

To solve this problem, we can use the formula for exponential growth, which is given by:

P(t) = P0 * e^(rt)

Where:
- P(t) is the population at time t
- P0 is the initial population
- r is the growth rate
- e is the base of the natural logarithm, approximately 2.71828

Given that the growth rate of the bacteria culture is 0.3, and the initial population is 40, we can plug these values into the formula to find the population at a specific time.

(1) Population after 30 days:
To find the population after 30 days, we need to substitute t = 30 into the formula and calculate P(30).

P(30) = 40 * e^(0.3 * 30)
= 40 * e^9

Using a calculator or a math software, we can find the approximate value of e^9, which is about 8103.08.

P(30) ≈ 40 * 8103.08
≈ 324123.2

Therefore, the population after 30 days is approximately 324123.

(2) Time to double the population:
To find the time it takes for the population to double, we need to solve the equation P(t) = 2P0, where P0 is the initial population.

2P0 = P0 * e^(rt)

We can divide both sides of the equation by P0 and substitute r = 0.3 to solve for t:

2 = e^(0.3t)

To isolate t, we take the natural logarithm of both sides:

ln(2) = 0.3t

Next, we divide both sides by 0.3:

t = ln(2) / 0.3

Using a calculator, we can find that ln(2)/0.3 is approximately 2.31.

Therefore, it takes approximately 2.31 days for the population to double.

This is really the same as your other question

p = 40 e^.3t

p(30) = 40 e^9

p(30) = 324,123
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2 = e^.3t
ln 2 = .693 = .3t
t = 2.3 days