The population of a town is 500,000. If the population growth rate is 6% yearly, when will the population exceed 600,000? (Express the answer correct to the nearest integer.)A. 2B. 3C. 4D. 5
Q. The population of a town is 500,000. If the population growth rate is 6% yearly, when will the population exceed 600,000? (Express the answer correct to the nearest integer.)A. 2B. 3C. 4D. 5
Identify Population and Growth Rate: Identify the initial population and the growth rate.The initial population P0 is 500,000 and the growth rate r is 6% per year.
Set Up Exponential Growth Formula: Set up the exponential growth formula.The formula for exponential growth is P(t)=P0×(1+r)t, where P(t) is the population at time t, P0 is the initial population, r is the growth rate, and t is the time in years.
Determine Population Threshold: Determine the population at which we want to find the time. We want to find the time when the population exceeds 600,000.
Set Up Inequality for t: Set up the inequality to solve for t.600,000<500,000×(1+0.06)t
Isolate Exponential Expression: Divide both sides of the inequality by 500,000 to isolate the exponential expression.500,000600,000<(1+0.06)t1.2<(1.06)t
Use Logarithms to Solve: Use logarithms to solve for t. Take the natural logarithm (ln) of both sides to get: ln(1.2)<t×ln(1.06)
Divide by ln(1.06) for t: Divide both sides by ln(1.06) to solve for t. t>ln(1.06)ln(1.2)
Calculate t Value: Calculate the value of t using a calculator.t>ln(1.06)ln(1.2)t>0.058268908123975340.1823215567939546t>3.129496404455733
Round t to Nearest Integer: Round t to the nearest integer since we cannot have a fraction of a year.t≈4 (since the population must exceed 600,000, we round up to the next whole year)
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