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To solve this problem, we are asked to find the rate of change in the world's population for the year 2013. The population model is given as:

\[
P(t) = 5.3e^{0.02t}
\]

where \( P(t) \) is the world's population in billions, and \( t \) represents the number of years since 1990.

### Step 1: Find \( t \) for the year 2013
Since \( t \) is the number of years since 1990, we can calculate \( t \) for the year 2013 as:

\[
t = 2013 - 1990 = 23 \, \text{years}
\]

### Step 2: Compute the rate of change (derivative)
To find the rate of change in the population, we need to compute the derivative of \( P(t) \) with respect to time \( t \). The derivative of the function \( P(t) = 5.3e^{0.02t} \) is given by:

\[
P'(t) = \frac{d}{dt} \left( 5.3e^{0.02t} \right)
\]

Using the chain rule:

\[
P'(t) = 5.3 \times 0.02 \times e^{0.02t} = 0.106e^{0.02t}
\]

### Step 3: Evaluate \( P'(t) \) at \( t = 23 \)
Now, we substitute \( t = 23 \) into the derivative:

\[
P'(23) = 0.106e^{0.02 \times 23} = 0.106e^{0.46}
\]

Using \( e^{0.46} \approx 1.584 \):

\[
P'(23) \approx 0.106 \times 1.584 \approx 0.1679 \, \text{billion people per year}
\]

Since \( 0.1679 \, \text{billion people} = 167,900,000 \, \text{people per year} \), the closest option is **168,000,000 people per year**.

### Answer: 168,000,000 people per year.

Would you like more details or have any questions?

Here are 5 related questions:
1. What assumptions are made by using an exponential model for population growth?
2. How would the result change if we had a different growth rate?
3. How can we interpret the derivative \( P'(t) \) in real-world terms?
4. What would the world population be in the year 2025 using this model?
5. How can this model be modified to account for changing growth rates over time?

**Tip:** The derivative of a population model helps to understand how fast the population is growing at a particular moment in time.

According to the United Nations data, the world population at the beginning of 1990 can be approximated by the function P(t) = 5.3e^(0.02t), where P(t) is the world's population in billions and t is the number of years since 1990. What is the rate of change in the world's population in the year 2013?

Solve the exponential population growth problem using calculus. Learn how to find the rate of change of the world's population in 2013 with detailed step-by-step explanations. This problem involves calculating derivatives and interpreting the results.

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