If a bacteria population doubles in 7 days, when was it half of its present population?
To determine when the bacteria population was half of its present population, we can use the concept of exponential growth.
If the bacteria population doubles every 7 days, it means that the growth rate is constant over time. This can be expressed mathematically using the formula:
N = N₀ * 2^(t/g)
where:
- N is the final population size
- N₀ is the initial population size
- t is the time passed
- g is the time it takes for the population to double
In this case, we know that the bacteria population doubles every 7 days, so g = 7. Let's assume the present population is N.
To find when the bacteria population was half of its present population (N/2), we can rearrange the formula as follows:
N/2 = N₀ * 2^(t/7)
Now we need to solve for t. Divide both sides of the equation by N₀ and take the logarithm (base 2) of both sides to isolate t:
log₂(N/2) = log₂(N₀ * 2^(t/7))
log₂(N/2) = log₂(N₀) + (t/7) * log₂(2)
log₂(N/2) - log₂(N₀) = (t/7) * log₂(2)
log₂(N/2N₀) = (t/7)
Finally, we multiply both sides by 7 to solve for t:
t = 7 * log₂(N/2N₀)
By substituting the known values (N = present population, N₀ = initial population), you can calculate the value of t in days.