To solve this problem, we need to follow the steps to find the area of a trapezoid using the provided formula [tex]\( A = \frac{1}{2} (b_1 + b_2) h \)[/tex]. Let's work through this step-by-step.
1. Identify the given values:
- The length of the first base ([tex]\(b_1\)[/tex]) is 3.6 cm.
- The length of the second base ([tex]\(b_2\)[/tex]) is [tex]\( 12 \frac{1}{3} \)[/tex] cm. Converting this mixed number to an improper fraction, we have:
[tex]\[
12 \frac{1}{3} = 12 + \frac{1}{3} = 12.3333 \text{ cm approximately}.
\][/tex]
- The height ([tex]\(h\)[/tex]) of the trapezoid is [tex]\(\sqrt{5} \, \text{cm}\)[/tex].
2. Calculate the sum of the bases:
[tex]\[
b_1 + b_2 = 3.6 + 12.3333 = 15.9333 \text{ cm}.
\][/tex]
3. Calculate the area using the formula:
[tex]\[
A = \frac{1}{2} (b_1 + b_2) h.
\][/tex]
Substituting the values:
[tex]\[
A = \frac{1}{2} (15.9333) \sqrt{5}.
\][/tex]
Since [tex]\( \sqrt{5} \)[/tex] is approximately 2.2361, the final step involves computing:
[tex]\[
A = \frac{1}{2} \times 15.9333 \times 2.2361 \approx 17.8140 \, \text{cm}^2.
\][/tex]
4. Reason for the area being irrational:
- The height ([tex]\(\sqrt{5}\)[/tex]) is an irrational number.
- When this irrational number (the height) is multiplied by rational numbers (the bases' sum), the resulting product is irrational.
Thus, the correct justification for the area being irrational is:
- The height is irrational, and it is multiplied by the other rational dimensions.
In conclusion, the area of the trapezoid is approximately [tex]\( 17.8140 \)[/tex] square centimeters, and this area is irrational because the height ([tex]\(\sqrt{5}\)[/tex]) is irrational, and it is multiplied by the other rational dimensions (the bases' sum).
