A bacteria culture is started with 300 bacteria. After 4 hours, the population has
grown to 500 bacteria. If the population grows exponentially,
a. Write a recursive formula for the number of bacteria
b. Write an explicit formula for the number of bacteria
c. If this trend continues, how many bacteria will there be in 1 day?
d. How long does it take for the culture to triple in size?

We are given that a bacteria culture starts with 300 bacteria and grows exponentially. After 4 hours, the population has grown to 500. We will solve this step-by-step.

### Step 1: Understanding the general formula for exponential growth
In exponential growth, the population at any time \( t \) (in hours) is given by the equation:
\[
P(t) = P_0 \cdot e^{kt}
\]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population (300 in this case),
- \( k \) is the growth rate constant,
- \( t \) is the time (in hours),
- \( e \) is the base of the natural logarithm.

### Part (a): Recursive formula for the number of bacteria

The recursive formula is based on the idea that the bacteria population at time \( t+1 \) is related to the population at time \( t \) by a constant growth factor.

Let \( P(t) \) represent the population at time \( t \). Then, the recursive formula can be written as:
\[
P(t+1) = P(t) \cdot r
\]
where \( r \) is the growth factor, which can be found by comparing the population after 4 hours to the initial population.

To find \( r \), we calculate the growth over 4 hours:
\[
P(4) = P_0 \cdot r^4 = 500
\]
\[
300 \cdot r^4 = 500
\]
\[
r^4 = \frac{500}{300} \approx 1.6667
\]
\[
r = \sqrt[4]{1.6667} \approx 1.137
\]
Thus, the recursive formula is:
\[
P(t+1) = P(t) \cdot 1.137
\]
with \( P(0) = 300 \).

### Part (b): Explicit formula for the number of bacteria

Using the general form for exponential growth, we already know the equation:
\[
P(t) = P_0 \cdot e^{kt}
\]
We need to find \( k \). To do this, we use the information that after 4 hours, the population is 500:
\[
500 = 300 \cdot e^{4k}
\]
Solving for \( k \):
\[
\frac{500}{300} = e^{4k}
\]
\[
\ln\left(\frac{5}{3}\right) = 4k
\]
\[
k = \frac{1}{4} \ln\left(\frac{5}{3}\right) \approx 0.1017
\]

Thus, the explicit formula is:
\[
P(t) = 300 \cdot e^{0.1017t}
\]

### Part (c): Number of bacteria after 1 day (24 hours)

We can now use the explicit formula to calculate the population after 24 hours:
\[
P(24) = 300 \cdot e^{0.1017 \cdot 24}
\]
\[
P(24) = 300 \cdot e^{2.4408} \approx 300 \cdot 11.477 \approx 3443
\]
So, there will be approximately **3,443 bacteria** after 1 day.

### Part (d): Time to triple the population

To find how long it takes for the population to triple, we need to solve for \( t \) when the population is 3 times the initial value:
\[
P(t) = 3 \cdot P_0 = 900
\]
Using the explicit formula:
\[
900 = 300 \cdot e^{0.1017t}
\]
\[
3 = e^{0.1017t}
\]
Taking the natural logarithm on both sides:
\[
\ln(3) = 0.1017t
\]
\[
t = \frac{\ln(3)}{0.1017} \approx 10.77
\]
So, it will take approximately **10.77 hours** for the population to triple in size.

---

Would you like more details or explanations on any of these steps?

Here are five questions to expand your understanding:

1. How do you calculate the growth rate \( r \) in the recursive formula for exponential growth?
2. What are the key differences between recursive and explicit formulas?
3. Why do we use natural logarithms when solving for the time in exponential equations?
4. How would the solution change if the bacteria grew logarithmically instead of exponentially?
5. How does the time to triple the population change if the initial population is different?

**Tip:** Always check if the growth rate (or decay rate) makes sense by verifying the result over a small time period.

A bacteria culture is started with 300 bacteria. After 4 hours, the population has grown to 500 bacteria. If the population grows exponentially: a. Write a recursive formula for the number of bacteria b. Write an explicit formula for the number of bacteria c. If this trend continues, how many bacteria will there be in 1 day? d. How long does it take for the culture to triple in size?

Solve the problem of a bacteria culture growing exponentially from 300 to 500 in 4 hours. Learn how to write recursive and explicit formulas, calculate the population after 1 day, and determine how long it takes for the culture to triple in size. This step-by-step solution covers key exponential growth concepts and logarithmic equations. Suitable for students in Grades 10-12.

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