A population of 7000 fruit flies is dying off at a rate of 8% per minute. How long will it take for only 700 fruit flies to remain alive?
After t minutes, there are 0.92^t of the original population alive. So, just solve
7000 * 0.92^t = 700
0.92^t = 0.1
t log 0.92 = log 0.1
t = 27.6
Thank you!
To calculate how long it will take for only 700 fruit flies to remain alive, we need to find the time it takes for the population to decrease from 7000 to 700.
To do this, we can use exponential decay formula:
N(t) = N₀ * e^(rt),
Where:
N(t) is the population at time t,
N₀ is the initial population,
e is the base of the natural logarithm (approximately 2.718),
r is the decay rate as a decimal,
t is the time in minutes.
We are given that N₀ = 7000, N(t) = 700, and the decay rate r = 8% = 0.08.
Substituting these values into the formula, we get:
700 = 7000 * e^(0.08t).
To solve for t, we need to isolate it on one side of the equation. Let's divide both sides of the equation by 7000:
700/7000 = e^(0.08t).
Simplifying further:
1/10 = e^(0.08t).
To solve for t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(1/10) = ln(e^(0.08t)).
Applying the logarithm property, ln(ab) = b * ln(a), we have:
ln(1/10) = 0.08t * ln(e).
Since ln(e) is 1, the equation simplifies to:
ln(1/10) = 0.08t.
Using a calculator or logarithm table, we find that ln(1/10) is approximately -2.3026. Substituting this value into the equation:
-2.3026 = 0.08t.
Now to solve for t, divide both sides by 0.08:
-2.3026/0.08 = t.
Calculating this gives us:
t ≈ -28.78.
The negative value for t doesn't make sense in this context since time cannot be negative. Therefore, we ignore the negative sign.
Hence, it will take approximately 28.78 minutes for only 700 fruit flies to remain alive.