Questions: If the population continues growing at the same rate, how many years will it take to reach 6 million? Round to the nearest tenth. Do not round intermediate steps. It will take approximately years for the population to reach 6 million.

If the population continues growing at the same rate, how many years will it take to reach 6 million? Round to the nearest tenth. Do not round intermediate steps.

It will take approximately years for the population to reach 6 million.

Transcript text: If the population continues growing at the same rate, how many years will it take to reach 6 million? Round to the nearest tenth. Do not rou intermediate steps. It will take approximately $\square$ years for the population to reach 6 million. $\square$

Solution

Solution Steps

To solve this problem, we need to determine the number of years it will take for the population to reach 6 million given a constant growth rate. We can use the formula for exponential growth: $ P(t) = P_0 \times e^{rt} $, where $ P(t) $ is the future population, $ P_0 $ is the initial population, $ r $ is the growth rate, and $ t $ is the time in years. We will solve for $ t $.

Step 1: Define the Variables

Let $ P_0 = 5,000,000 $ be the initial population, $ P = 6,000,000 $ be the target population, and $ r = 0.02 $ be the growth rate.

Step 2: Use the Exponential Growth Formula

We apply the formula for exponential growth: \[ P(t) = P_0 \times e^{rt} \] To find $ t $, we rearrange the formula: \[ t = \frac{\ln(P) - \ln(P_0)}{r} \]

Step 3: Calculate $ t $

Substituting the values into the equation: \[ t = \frac{\ln(6,000,000) - \ln(5,000,000)}{0.02} \] Calculating this gives: \[ t \approx 9.1161 \] Rounding to the nearest tenth, we find: \[ t \approx 9.1 \]