Question

The population of Kenya reached 25,000,000 people in 1990.
When will it reach 50,000,000 people? Assume an annual rate of increase of
4,1%.

Answer 1

Answer:

In 2007

Step-by-step explanation:

To find out when the population of Kenya will reach 50,000,000 people with an annual rate of increase of 4.1%, we can use the exponential growth formula:

$P(t) = P_0 \times (1 + r)^t$

Where:

• $P(t)$ is the population at time $t$.
• $P_0$ is the initial population (in 1990 in this case).
• $r$ is the annual rate of increase (in decimal form).
• $t$ is the time in years.

In this case, $P_0 = 25,000,000$, $r = 0.041$ (4.1% expressed as a decimal), and we want to find $t$ when $P(t) = 50,000,000$.

So, we set up the equation:

$50,000,000 = 25,000,000 \times (1 + 0.041)^t$

Now, we need to solve for $t$.

Start by dividing both sides by $25,000,000$:

$2 = (1 + 0.041)^t$

Now, take the natural logarithm (ln) of both sides:

$\ln(2) = \ln\left((1 + 0.041)^t\right)$

Using the property of logarithms that $\ln(a^b) = b \cdot \ln(a)$, we have:

$\ln(2) = t \cdot \ln(1 + 0.041)$

Now, solve for $t$:

$t = \dfrac{\ln(2)}{\ln(1 + 0.041)}$

$t = \dfrac{\ln(2)}{\ln(1.041)}$

Using a calculator:

$t \approx \dfrac{0.6931471806}{0.04018178963}$

$t\approx 17.25028146$

$t \approx 17.25 \textsf{(in 2 d.p.)}$

So, it will take approximately 17.25 years from 1990 for the population of Kenya to reach 50,000,000 people with an annual rate of increase of 4.1%.

Since we can't have a fraction of a year in this context, we round up to the nearest whole number.

Therefore, the population is projected to reach 50,000,000 people around the year [tex]1990 + 17 = 2007[/tex].