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Rats have recently become a nuisance in Norwood. Efforts to control the rat problem have led to the rat population declining by $15\%$ every year. If the town currently has $13,000$ rats, how many rats will there be in $3$ years? $\newline$ If necessary, round your answer to the nearest whole number. $\newline$ ____ rats $\newline$

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Exponential growth and decay: word problems

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Q. Rats have recently become a nuisance in Norwood. Efforts to control the rat problem have led to the rat population declining by

15\%

every year. If the town currently has

13,000

rats, how many rats will there be in

3

years?

\newline

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

\newline

____ rats

\newline

Identify Population and Decline Rate: Identify the initial population and the rate of decline. $\newline$ The initial population of rats is $13,000$ , and the population declines by $15\%$ every year.
Determine Decay Factor: Determine the decay factor. $\newline$ Since the population declines by $15\%$ , the decay factor is $1 - 0.15 = 0.85$ .
Apply Exponential Decay Formula: Apply the exponential decay formula. $\newline$ The formula for exponential decay is $P(t) = P_0 \times (\text{decay factor})^{t}$ , where $P(t)$ is the population at time $t$ , $P_0$ is the initial population, and $t$ is the number of years.
Calculate Population After $3$ Years: Calculate the population after $3$ years. $P(3) = 13,000 \times (0.85)^{3}$
Perform Calculation: Perform the calculation. $\newline$ $P(3) = 13,000 \times (0.85)^{3}$ $\newline$ $P(3) = 13,000 \times 0.85 \times 0.85 \times 0.85$ $\newline$ $P(3) = 13,000 \times 0.614125$ $\newline$ $P(3) = 7,983.625$
Round to Nearest Whole Number: Round the answer to the nearest whole number. $\newline$ The population of rats after $3$ years will be approximately $7,984$ rats.

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