A certain population of bacteria doubles every 3 weeks. How long will it take for a population of 3,500 to triple?

I assume you are studying exponential equations and you also know how to work with logs.

your equation would be
N = a(2)^(t/3), were t is in weeks, a is the initial number of bacteria, and N is the number of bacteria you are after.

so 3a = a(2)^(t/3)
3 = (2)^(t/3)
t/3 = log3/log2
= 1.58496
t = 3(1.58496)
= 4.75

it would take 4.75 weeks to triple

To solve this problem, we need to determine the number of weeks it takes for a population of bacteria to triple.

Let's assume that the initial population is denoted as P₀, and the population after a certain number of weeks is denoted as P.

According to the information given, the population doubles every 3 weeks. Therefore, the population is proportional to 2 raised to the power of the number of 3 week intervals that have passed.

We can express this mathematically as:
P = P₀ * 2^(n/3)

For the current situation, we have:
P₀ = 3,500 (initial population)
P = 3 * P₀ (triple the initial population)

Substituting these values into the equation, we get:
3 * P₀ = P₀ * 2^(n/3)

Now, let's solve for n, the number of 3-week intervals it takes for the population to triple:

3 = 2^(n/3)

To remove the exponent, we can take the logarithm (base 2) of both sides:

log₂(3) = n/3 * log₂(2)

Using properties of logarithms, we can simplify this equation to:

log₂(3) = n/3

Now, let's solve for n:

n = 3 * log₂(3)

Using a calculator, we find that log₂(3) is approximately 1.585.

n = 3 * 1.585
n ≈ 4.755

Since n represents the number of 3-week intervals, we can round it up to the nearest whole number to get the smallest number of intervals required for the population to triple.

Therefore, it will take approximately 5 intervals of 3 weeks for the population of bacteria to triple.

To determine the total time in weeks, we multiply the number of intervals by 3 weeks:

Total Time = 5 * 3
Total Time = 15 weeks

Therefore, it will take approximately 15 weeks for a population of 3,500 bacteria to triple.

To determine how long it will take for a population of bacteria to triple, we need to calculate the number of times the population needs to double. Let's break down the problem:

1) Find the initial number of bacteria: The initial population is given as 3,500.

2) Calculate the number of times the population doubles: A population doubles every 3 weeks. To find the number of times it will double to triple, we divide the tripling factor (3) by the doubling factor (2). In this case, 3 ÷ 2 equals 1.5.

3) Determine the time it takes for the population to triple: Since each doubling occurs every 3 weeks, we multiply the number of times the population doubles (1.5) by the doubling period (3 weeks). This will give us the total time it takes for the population to triple.

Let's calculate it step by step:

Step 1: Initial population = 3,500
Step 2: Number of times population doubles: 3 ÷ 2 = 1.5
Step 3: Total time to triple = Number of times doubles × Doubling period = 1.5 × 3 weeks = 4.5 weeks

Therefore, it will take approximately 4.5 weeks for the population of bacteria to triple.