Answer:
- f(t) = 10000·0.91^5
- about 7.3 years
Explanation:
You want the formula that describes the decay of an insect population from 10000 by 9% per year, and the years it takes for the population to decline by half.
Exponential function
An exponential function will have the form ...
f(t) = (initial value) · (growth factor)^t
where ...
growth factor = 1 + growth rate
Application
Here, the growth rate is -9% per year, so the growth factor is ...
1 -9% = 1 -0.09 = 0.91
The population is then ...
f(t) = 10000·0.91^t . . . . . formula for number of insects
Half life
We can solve for t when f(t) = 5000 (half the original population):
5000 = 10000·0.91^t
1/2 = 0.91^t . . . . . . . . . . . divide by 10,000
ln(1/2) = t·ln(0.91) . . . . . take logarithms
t = ln(0.5)/ln(0.91) ≈ 7.3496 ≈ 7.3 . . . . years
It would take about 7.3 years for the population to reduce to less than half its present size.
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Additional comment
We have rounded the time period to the nearest tenth, 7.3 years. At the end of that period, the population is modeled to be about 5023, not quite "less than half." To get below half the original population would take almost 7.35 years.
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