the growth of the worlds population can be described by the equation a = a₀eʳᵗ, where time t is measure in…

the growth of the worlds population can be described by the equation a = a₀eʳᵗ, where time t is measure in years, a₀ 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 = 3% per year. how long will it take a population of 2 billion to increase to 3 billion. years (round to the nearest whole number.)

Answer

Explanation:

Step1: Substitute values into formula

Given $A = 3$ billion, $A_0=2$ billion, $r = 0.03$. Substitute into $A = A_0e^{rt}$: $3=2e^{0.03t}$.

Step2: Isolate the exponential term

Divide both sides by 2: $\frac{3}{2}=e^{0.03t}$.

Step3: Take natural - logarithm of both sides

$\ln(\frac{3}{2})=\ln(e^{0.03t})$. Since $\ln(e^{x}) = x$, we have $\ln(\frac{3}{2}) = 0.03t$.

Step4: Solve for t

$t=\frac{\ln(\frac{3}{2})}{0.03}$. Calculate $\ln(\frac{3}{2})\approx0.4055$, then $t=\frac{0.4055}{0.03}\approx13.52$.