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A town has a population of 
1.025 ×10^(5) and shrinks at a rate of 
5% every year. Which equation represents the town's population after 6 years?

P=(1.025 ×10^(5))(1-0.05)(1-0.05)(1-0.05)

P=(1.025 ×10^(5))(0.05)^(6)

P=(1.025 ×10^(5))(0.95)^(6)

P=(1.025 ×10^(5))(1-0.5)^(6)

A town has a population of 1.025×105 1.025 \times 10^{5} and shrinks at a rate of 5% 5 \% every year. Which equation represents the town's population after 66 years?\newlineP=(1.025×105)(10.05)(10.05)(10.05) P=\left(1.025 \times 10^{5}\right)(1-0.05)(1-0.05)(1-0.05) \newlineP=(1.025×105)(0.05)6 P=\left(1.025 \times 10^{5}\right)(0.05)^{6} \newlineP=(1.025×105)(0.95)6 P=\left(1.025 \times 10^{5}\right)(0.95)^{6} \newlineP=(1.025×105)(10.5)6 P=\left(1.025 \times 10^{5}\right)(1-0.5)^{6}

Full solution

Q. A town has a population of 1.025×105 1.025 \times 10^{5} and shrinks at a rate of 5% 5 \% every year. Which equation represents the town's population after 66 years?\newlineP=(1.025×105)(10.05)(10.05)(10.05) P=\left(1.025 \times 10^{5}\right)(1-0.05)(1-0.05)(1-0.05) \newlineP=(1.025×105)(0.05)6 P=\left(1.025 \times 10^{5}\right)(0.05)^{6} \newlineP=(1.025×105)(0.95)6 P=\left(1.025 \times 10^{5}\right)(0.95)^{6} \newlineP=(1.025×105)(10.5)6 P=\left(1.025 \times 10^{5}\right)(1-0.5)^{6}
  1. Identify Population and Rate: Identify the initial population and the annual shrink rate.\newlineThe initial population is given as 1.025×1051.025 \times 10^5, and the town shrinks at a rate of 55% every year, which means the population is multiplied by 9595% (or 00.9595) each year.
  2. Determine Formula for Population: Determine the correct formula to represent the population after 66 years.\newlineSince the population decreases by a constant percentage each year, this is an exponential decay problem. The general formula for exponential decay is P(t)=P0×(1r)tP(t) = P_0 \times (1 - r)^t, where P0P_0 is the initial population, rr is the decay rate, and tt is the time in years.
  3. Substitute Values into Formula: Substitute the given values into the exponential decay formula.\newlineThe initial population P0P_0 is 1.025×1051.025 \times 10^5, the decay rate rr is 00.0505 (55%), and the time tt is 66 years. Plugging these values into the formula gives us P(6)=(1.025×105)×(10.05)6P(6) = (1.025 \times 10^5) \times (1 - 0.05)^6.
  4. Simplify Equation: Simplify the equation to find the correct answer.\newlineThe correct equation after simplifying is P(6)=(1.025×105)×(0.95)6P(6) = (1.025 \times 10^5) \times (0.95)^6, which matches one of the provided options.

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