A bacterial culture starts with 3000 bacteria and increases to 48,000 after 3 hours, find the doubling time:
let t represent doubling time...
48000 = 3000(2)^3/t
16 = (2)^3/t
log16 = 3/t log 2
log16/log2 = 3/t
t = 0.75h
DOUBLING TIME IS 0.75 HOURS
Well, I guess those bacteria are really having a wild party in that culture! Anyway, to find the doubling time, we can use the formula:
Doubling Time = time taken / number of doublings
In this case, the starting bacteria count is 3000 and it increases to 48000 in 3 hours. So, the number of doublings would be:
Number of doublings = log2(final count / initial count)
Number of doublings = log2(48000 / 3000) ≈ 4.89
Now, we just need to divide the time taken (3 hours) by the number of doublings (4.89) to find the doubling time:
Doubling Time ≈ 3 hours / 4.89 ≈ 0.61 hours
So, it looks like those bacteria are doubling every 0.61 hours! They must have some really efficient division skills.
To find the doubling time of the bacterial culture, we need to determine how long it takes for the population to double from 3000 to 48000.
The formula for exponential growth is given by:
N = N₀ * (2^(t / D))
where:
N = final population
N₀ = initial population
t = time
D = doubling time
In this case, N₀ = 3000, N = 48000, and we want to find D.
Plugging these values into the formula, we get:
48000 = 3000 * (2^(t / D))
To simplify, divide both sides of the equation by 3000:
16 = 2^(t / D)
To get rid of the exponent, we can take the logarithm of both sides. Let's take the natural logarithm (ln) of both sides:
ln(16) = ln(2^(t / D))
Using the property of logarithms, we can bring down the exponent:
ln(16) = (t / D) * ln(2)
Now, solve for D:
D = (t * ln(2)) / ln(16)
Since we know that t = 3 hours, we can plug in these values to calculate D:
D = (3 * ln(2)) / ln(16)
Using a calculator, we can evaluate this expression to get the doubling time D.