Question

1. a population starts at 2000 and increases at a rate of 4% per year. what is the population after 5 years?

a. 3531
b. 2552
c. 1631
d. 2433

Explanation:

Step1: Recall the compound growth formula

The formula for compound growth is $P = P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the annual growth rate (as a decimal), and $t$ is the number of years.

Step2: Identify the values

Here, $P_0 = 2000$, $r = 4\% = 0.04$, and $t = 5$.

Step3: Substitute into the formula

Substitute the values into the formula: $P = 2000(1 + 0.04)^5$.

Step4: Calculate $(1 + 0.04)^5$

First, calculate $1 + 0.04 = 1.04$. Then, $1.04^5 \approx 1.2166529024$.

Step5: Calculate the population

Multiply by the initial population: $P = 2000 \times 1.2166529024 \approx 2433.3058048$. Rounding to the nearest whole number, we get approximately 2433. Wait, but let's check again. Wait, maybe I made a mistake. Wait, 1.04^5: 1.04^2 = 1.0816, 1.04^3 = 1.08161.04 = 1.124864, 1.04^4 = 1.1248641.04 = 1.16985856, 1.04^5 = 1.169858561.04 = 1.2166529024. Then 20001.2166529024 = 2433.3058..., which is approximately 2433. But wait, the option b is 2552, d is 2433. Wait, maybe I miscalculated? Wait, no, let's check again. Wait, 4% growth per year, 5 years. Let's do it step by step:

Year 1: 2000*1.04 = 2080

Year 2: 2080*1.04 = 2163.2

Year 3: 2163.2*1.04 = 2249.728

Year 4: 2249.728*1.04 = 2339.71712

Year 5: 2339.71712*1.04 = 2433.3058048. Yes, so it's approximately 2433, which is option d. Wait, but the initial calculation gave 2433, which is option d. Wait, but let's check the options again. The options are a. 3531, b. 2552, c. 1631, d. 2433. So the correct answer should be d. Wait, but maybe I made a mistake in the formula? No, the compound growth formula is correct for exponential growth, which is appropriate for population growth (assuming continuous or annual compounding, here it's annual). So the calculation shows that after 5 years, the population is approximately 2433, which is option d.

Answer:

d. 2433