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A little-known species of insect is approaching extinction, with a population that falls by $10\%$ every year. There are currently $2,400$ insects remaining. How many will there be in $4$ years? If necessary, round your answer to the nearest whole number. $\newline$ $\newline$ $\_\_\_\_$ insects $\newline$

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Exponential growth and decay: word problems

Full solution

Q. A little-known species of insect is approaching extinction, with a population that falls by

10\%

every year. There are currently

2,400

insects remaining. How many will there be in

4

years? If necessary, round your answer to the nearest whole number.

\newline

\newline

\_\_\_\_

insects

\newline

Identify Population and Rate: Identify the initial population and the rate of decrease. The initial population $P_0$ is $2,400$ insects, and the rate of decrease is $10\%$ per year.
Formula for Decay: Determine the formula for exponential decay. The formula for exponential decay is $P(t) = P_0 \times (1 - r)^t$ , where $P(t)$ is the population at time $t$ , $P_0$ is the initial population, $r$ is the rate of decrease (expressed as a decimal), and $t$ is the time in years.
Convert Rate to Decimal: Convert the rate of decrease to a decimal. $\newline$ The rate of decrease is $10\%$ , which as a decimal is $0.10$ .
Calculate Population After $4$ Years: Calculate the population after $4$ years. Using the formula $P(t) = P_0 \times (1 - r)^t$ , we substitute $P_0 = 2,400$ , $r = 0.10$ , and $t = 4$ . $P(4) = 2,400 \times (1 - 0.10)^4$
Perform Calculation: Perform the calculation. $\newline$ $P(4) = 2,400 \times (0.90)^4$ $\newline$ $P(4) = 2,400 \times 0.6561$ (rounded to four decimal places) $\newline$ $P(4) = 1,574.64$
Round to Nearest Whole Number: Round the answer to the nearest whole number. $\newline$ The population after $4$ years, rounded to the nearest whole number, is $1,575$ insects.

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