QuestionJuly 23, 2025

Suppose a population of fruit flies increases at a rate of g(x)=3e^0.09t flies per day. If the population starts with 46 flies how many flies will there be after 11 days? square (Round down to a whole number.)

Suppose a population of fruit flies increases at a rate of g(x)=3e^0.09t flies per day. If the population starts with 46 flies how many flies will there be after 11 days? square (Round down to a whole number.)
Suppose a population of fruit flies increases at a rate of g(x)=3e^0.09t flies per day. If the population starts with 46 flies how many flies will there be after 11 days? square (Round down to a whole number.)

Solution
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Answer

139 flies Explanation 1. Integrate the growth rate function Integrate g(t) = 3e^{0.09t} with respect to t to find the total increase in population over time. The integral is \int 3e^{0.09t} \, dt = \frac{3}{0.09} e^{0.09t} + C = 33.33e^{0.09t} + C. 2. Calculate the constant of integration Use the initial condition that at t=0, the population is 46 flies. Thus, 33.33e^{0.09(0)} + C = 46. Solving gives C = 46 - 33.33 = 12.67. 3. Determine the population after 11 days Substitute t = 11 into the integrated function: P(11) = 33.33e^{0.09 \times 11} + 12.67. Calculate this value.

Explanation

1. Integrate the growth rate function<br /> Integrate $g(t) = 3e^{0.09t}$ with respect to $t$ to find the total increase in population over time. The integral is $\int 3e^{0.09t} \, dt = \frac{3}{0.09} e^{0.09t} + C = 33.33e^{0.09t} + C$.<br />2. Calculate the constant of integration<br /> Use the initial condition that at $t=0$, the population is 46 flies. Thus, $33.33e^{0.09(0)} + C = 46$. Solving gives $C = 46 - 33.33 = 12.67$.<br />3. Determine the population after 11 days<br /> Substitute $t = 11$ into the integrated function: $P(11) = 33.33e^{0.09 \times 11} + 12.67$. Calculate this value.
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