Answer
$230.26$ days
Work Step by Step
Fly Population Growth Analysis Step 1 To solve this problem, first use Formula (13) to set up the equation for the number of flies in the container after $t$ days, then solve the equation $y(t) = 10000$. Step 2 Initially, there are 100 flies in the container, and the growth rate is 0.02. Since the growth is exponential, by Formula (13): \[ y(t) = 100e^{0.02t}, \] where $y(t)$ denotes the number of flies in the container after $t$ days. Step 3 Next, solve the equation $y(t) = 10000$: \[ 100e^{0.02t} = 10000. \] Divide by 100: \[ e^{0.02t} = 100. \] Take the natural logarithm: \[ 0.02t = \ln 100. \] Simplify: \[ t = \frac{\ln 100}{0.02} \approx 230.26. \] Step 4 Finally, we have calculated that the container will reach capacity after approximately 230 days. Result The container will reach capacity after approximately 230.26 days.
