Part (a) - Finding the population function $Q(t)$

We are given that the world population at the beginning of 1990 was 5.3 billion, and that it is growing at a constant rate of 2% per year. The general form for exponential growth is:

$Q(t) = Q_0 e^{rt}$

Where:

• $Q(t)$ is the population at time $t$ (in years),
• $Q_0$ is the initial population (at $t = 0$),
• $r$ is the growth rate (in decimal form),
• $t$ is the time in years.

For this problem:

• $Q_0 = 5.3$ billion (the population at the beginning of 1990),
• $r = 0.02$ (since the growth rate is 2% per year),
• $t = 0$ corresponds to the beginning of 1990.

So, the population function becomes:

$Q(t) = 5.3 e^{0.02t}$

Part (a) - Time required to double the population

To find the time $t_2$ it takes for the population to double, we need to solve for $t$ when $Q(t) = 2Q_0$. Since the initial population $Q_0$ is 5.3 billion, the population will double to $2 \times 5.3 = 10.6$ billion. Setting up the equation:

$10.6 = 5.3 e^{0.02t_2}$

Dividing both sides by 5.3:

$2 = e^{0.02t_2}$

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

$\ln(2) = 0.02t_2$

Solve for $t_2$:

$t_2 = \frac{\ln(2)}{0.02} \approx \frac{0.6931}{0.02} \approx 34.66 \text{ years}$

So, the population will double in approximately 34.66 years.

Part (b) - World population with a reduced growth rate of 1.3%

Now, if the growth rate is reduced to 1.3% (or $r = 0.013$), we can use the same exponential growth model but with the new rate. We already know that $t_2 = 34.66$ years from part (a), so we plug this into the modified equation to find the population after 34.66 years with a growth rate of 1.3%:

$Q(t_2) = 5.3 e^{0.013 \times 34.66}$

First, calculate the exponent:

$0.013 \times 34.66 \approx 0.4506$

Now compute the population:

$Q(34.66) = 5.3 e^{0.4506} \approx 5.3 \times 1.5699 \approx 8.31 \text{ billion}$

So, if the growth rate were reduced to 1.3% per year, the world population would be approximately 8.31 billion after 34.66 years.

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Would you like further explanation on any part of the solution?

Here are 5 related questions you might explore:

1. How would the formula change if the population growth rate was not constant but varying?
2. What would be the population after 50 years if the growth rate remains 2%?
3. Can exponential growth continue indefinitely? Why or why not?
4. What if the growth rate were negative, how would the function and results change?
5. How does the exponential growth model compare to logistic growth in terms of long-term predictions?

Tip: When dealing with exponential growth, always ensure that the growth rate is expressed as a decimal in the equation (e.g., 2% becomes 0.02).