A water sample in a laboratory initially contains 6000 bacteria. The organisms reproduce at a rate of 10% per hour. Find the function that corresponds to this situation. Then predict how long it will take for the population of bacteria to double in number. Round your answer to the nearest thousandth.
I see you aren't going to make any effort, but just repost till someone does it for you.
6000 * 1.1^t
now solve for when 1.1^t = 2
To find the function that corresponds to this situation, we can use the exponential growth formula:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at time t,
P0 is the initial population,
r is the growth rate as a decimal,
t is the time.
Given the following information:
P0 = 6000
r = 0.10 (10% can be written as 0.10)
The function becomes:
P(t) = 6000 * (1 + 0.10)^t
Now we need to predict how long it will take for the population to double in number. We can set P(t) to be twice the initial population and solve for t:
2 * P0 = P0 * (1 + r)^t
Simplifying the equation:
2 = (1 + 0.10)^t
Taking the logarithm of both sides to isolate the exponent:
log(2) = t * log(1.10)
Solving for t:
t = log(2) / log(1.10)
Using a calculator, we find:
t ≈ 6.931 / 0.041 = 169.146
Therefore, it will take approximately 169.146 hours for the population of bacteria to double in number. Rounded to the nearest thousandth, it would be 169.000 hours.
To find the function that corresponds to this situation, we can use the formula for exponential growth:
N(t) = N₀ * (1 + r)ˣ
Where:
N(t) is the population size at time t
N₀ is the initial population size
r is the growth rate per time period (expressed as a decimal)
x is the number of time periods
In this case, the initial population size is 6000, and the growth rate per hour is 10%, or 0.10 as a decimal. So we have:
N(t) = 6000 * (1 + 0.10)ˣ
To predict how long it will take for the population of bacteria to double in number, we can set N(t) equal to 2N₀ and solve for x:
2N₀ = N₀ * (1 + 0.10)ˣ
Dividing both sides of the equation by N₀, we get:
2 = 1.10ˣ
To solve for x, we can take the logarithm of both sides. Let's use the natural logarithm (ln) for this calculation:
ln(2) = ln(1.10ˣ)
Using the property of logarithms that states ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(2) = x * ln(1.10)
Now, solve for x by dividing both sides of the equation by ln(1.10):
x = ln(2) / ln(1.10)
Using a calculator to perform the division, we find:
x ≈ 6.93
Therefore, it will take approximately 6.93 hours for the population of bacteria to double in number.
Rounded to the nearest thousandth, the answer is approximately 6.930 hours.