To solve this problem, let's go through the steps of the calculation and analyze the final result.
1. Identify the given values:
- The lengths of the bases of the trapezoid are [tex]\( b_1 = 3.6 \)[/tex] cm and [tex]\( b_2 = 12 \frac{1}{3} \)[/tex] cm.
- The height of the trapezoid is [tex]\( h = \sqrt{5} \)[/tex] cm.
2. Convert mixed number to an improper fraction:
- The second base, [tex]\( b_2 = 12 \frac{1}{3} \)[/tex] can be converted to a decimal value:
[tex]\[
b_2 = 12 + \frac{1}{3} = 12 + 0.3333333333333333 \approx 12.333333333333334
\][/tex]
3. Calculate the sum of the bases:
- Add the lengths of the bases:
[tex]\[
\text{bases\_sum} = b_1 + b_2 = 3.6 + 12.333333333333334 \approx 15.933333333333334 \text{ cm}
\][/tex]
4. Calculate the area of the trapezoid:
- Use the formula for the area of a trapezoid:
[tex]\[
A = \frac{1}{2} \left( b_1 + b_2 \right) h
\][/tex]
- Substitute the values:
[tex]\[
A = \frac{1}{2} \times 15.933333333333334 \times \sqrt{5}
\][/tex]
- The height [tex]\( h = \sqrt{5} \approx 2.23606797749979 \)[/tex] (since [tex]\(\sqrt{5}\)[/tex] is an irrational number).
- Calculate the area:
[tex]\[
A \approx \frac{1}{2} \times 15.933333333333334 \times 2.23606797749979 \approx 17.814008220748327 \text{ square cm}
\][/tex]
5. Determine the rationality of the area:
- The area of the trapezoid is given by multiplying the sum of the bases (a rational number) with the height (an irrational number).
- When a rational number is multiplied by an irrational number, the result is an irrational number.
6. Explanation for the area being irrational:
- The area is irrational because the height [tex]\( \sqrt{5} \)[/tex] is irrational, and it is being multiplied by the rational sum of the bases.
In conclusion:
The area of the trapezoid is irrational because the height is irrational, and it is multiplied by the other rational dimensions.
