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To solve this problem using the trapezoidal rule, we need to estimate the integral of the function \( f(t) \) over the interval from \( t = 0 \) to \( t = 5 \). The given data points are:

\[ \begin{array}{c|cccccc}
t & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
f(t) & 54 & 55 & 56 & 63 & 52 & 48 \\
\end{array} \]

The trapezoidal rule formula for approximating the integral is given by:

\[ \int_a^b f(t) \, dt \approx \frac{b - a}{2n} \left[ f(t_0) + 2 \sum_{i=1}^{n-1} f(t_i) + f(t_n) \right] \]

Here, \( a = 0 \), \( b = 5 \), and \( n = 5 \) (since we have 6 points). Applying the trapezoidal rule:

\[ \int_0^5 f(t) \, dt \approx \frac{5 - 0}{2 \cdot 5} \left[ f(0) + 2(f(1) + f(2) + f(3) + f(4)) + f(5) \right] \]

\[ \int_0^5 f(t) \, dt \approx \frac{5}{10} \left[ 54 + 2(55 + 56 + 63 + 52) + 48 \right] \]

Calculate the sum inside the brackets:

\[ 54 + 2(55 + 56 + 63 + 52) + 48 = 54 + 2(226) + 48 \]
\[ = 54 + 452 + 48 \]
\[ = 554 \]

Now multiply by the factor outside the brackets:

\[ \int_0^5 f(t) \, dt \approx \frac{5}{10} \cdot 554 = 0.5 \cdot 554 = 277 \]

So, the estimated average income over the 5 years is \( 277 \) thousand dollars per year.

The correct answer is B) 277 nghìn đô-la/năm.

Do you have any questions or would you like more details?

Here are 5 additional questions you might want to ask:
1. How do you apply Simpson's rule for numerical integration?
2. What are the differences between the trapezoidal rule and Simpson's rule?
3. Can you explain the concept of numerical integration?
4. How would you approximate the integral if the function is given as a continuous function rather than discrete points?
5. What are some practical applications of numerical integration in real life?

**Tip:** When using numerical methods for integration, always ensure that the intervals are evenly spaced to apply the formulas correctly.

Learn how to estimate average income over 5 years using the trapezoidal rule for numerical integration. This problem involves applying the trapezoidal rule formula to discrete data points of income over time. Suitable for advanced high school students and undergraduates studying numerical methods.

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