A town has a population of 1.025×105 and shrinks at a rate of 5% every year. Which equation represents the town's population after 6 years?P=(1.025×105)(1−0.05)(1−0.05)(1−0.05)P=(1.025×105)(0.05)6P=(1.025×105)(0.95)6P=(1.025×105)(1−0.5)6
Q. A town has a population of 1.025×105 and shrinks at a rate of 5% every year. Which equation represents the town's population after 6 years?P=(1.025×105)(1−0.05)(1−0.05)(1−0.05)P=(1.025×105)(0.05)6P=(1.025×105)(0.95)6P=(1.025×105)(1−0.5)6
Identify Population and Rate: Identify the initial population and the annual shrink rate.The initial population is given as 1.025×105, and the town shrinks at a rate of 5% every year, which means the population is multiplied by 95% (or 0.95) each year.
Determine Formula for Population: Determine the correct formula to represent the population after 6 years.Since 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×(1−r)t, where P0 is the initial population, r is the decay rate, and t is the time in years.
Substitute Values into Formula: Substitute the given values into the exponential decay formula.The initial population P0 is 1.025×105, the decay rate r is 0.05 (5%), and the time t is 6 years. Plugging these values into the formula gives us P(6)=(1.025×105)×(1−0.05)6.
Simplify Equation: Simplify the equation to find the correct answer.The correct equation after simplifying is P(6)=(1.025×105)×(0.95)6, which matches one of the provided options.
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