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A population of 10 000 insects decreases by 9% every year. Write a formula to calculate the number of insects

left after a certain number of years. Use your formula to determine how many years it would take for the insect
population to reduce to less than half its present size.

User Muzzyq
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8.6k points

1 Answer

5 votes

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.

#95141404393

User Mlarsen
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