Ajay is researching how the population of his hometown has changed over time. Specifically, he learns his hometown had a population of 20,000 in 1990 , and that the population has since increased by about 8% every 3 years. Ajay predicts that his town can only support a population of 50,000 . Ajay is relieved to see that population has not exceeded 50,000 t years after 1990. Write an inequality in terms of t that models the situation.

Question

Ajay is researching how the population of his hometown has changed over time. Specifically, he learns his hometown had a population of 20,000 in 1990 , and that the population has since increased by about  8% every 3 years. Ajay predicts that his town can only support a population of 50,000 . Ajay is relieved to see that population has not exceeded  50,000 t years after 1990. Write an inequality in terms of  t that models the situation.

Bytelearn · Accepted Answer

Identify Population and Growth Rate: Identify the initial population and the growth rate. The initial population (P0) in 1990 is 20,000, and the growth rate is 8% every 3 years.Convert to Exponential Growth Factor: Convert the growth rate to an exponential growth factor. The growth rate of 8% every 3 years can be expressed as a growth factor (b) of 1 + 0.08 = 1.08. However, since the growth happens every 3 years, we need to adjust the exponent to reflect this. The population after t years can be modeled by the equation P(t) = P0 * b^(t/3), where t is the number of years after 1990.Write Population Inequality: Write the inequality that represents the population not exceeding 50,000. We want to ensure that the population P(t) does not exceed 50,000. Therefore, the inequality will be P(t) ≤ 50,000.Substitute Values into Inequality: Substitute the values of P0 and b into the inequality. Substituting the values we have, we get 20,000 * (1.08)^(t/3) ≤ 50,000.Simplify Final Inequality: Simplify the inequality to its final form. The inequality 20,000 * (1.08)^(t/3) ≤ 50,000 is already in its simplest form and correctly models the situation.

Full solution

More problems from Write exponential functions: word problems

Related topics