Question

the growth of the worlds population can be described by the equation $a = a_0e^{rt}$, where time $t$ is measure in years, $a_0$ is the population of the world at $t = 0$, $r$ is the annual growth rate, and $a$ is the population at time $t$. assume that $r = 4\\%$ per year. how long will it take a population of 10 billion to increase to 14 billion. \
ewline years (round to the nearest whole number.)

Explanation:

Step1: Identify known values

We know the population growth formula is \( A = A_0 e^{rt} \), where \( A_0 = 10 \) billion, \( A = 14 \) billion, and \( r = 4\% = 0.04 \) per year. We need to find \( t \).

Step2: Substitute values into the formula

Substitute \( A = 14 \), \( A_0 = 10 \), and \( r = 0.04 \) into \( A = A_0 e^{rt} \):

$$ 14 = 10 e^{0.04t} $$

Step3: Divide both sides by 10

$$ \frac{14}{10} = e^{0.04t} $$

$$ 1.4 = e^{0.04t} $$

Step4: Take the natural logarithm of both sides

$$ \ln(1.4) = \ln(e^{0.04t}) $$

Using the property \( \ln(e^x) = x \), we get:

$$ \ln(1.4) = 0.04t $$

Step5: Solve for \( t \)

$$ t = \frac{\ln(1.4)}{0.04} $$

Calculate \( \ln(1.4) \approx 0.3365 \), then:

$$ t \approx \frac{0.3365}{0.04} \approx 8.41 $$

Round to the nearest whole number, \( t \approx 8 \).

Answer:

8