To find the area of the trapezoid, we can use the formula for the area of a trapezoid:
[tex]\[ A = \frac{1}{2} (b_1 + b_2) h \][/tex]
Given the values:
- [tex]\( b_1 = 3.6 \)[/tex] cm
- [tex]\( b_2 = 12 \frac{1}{3} \)[/tex] cm
- [tex]\( h = \sqrt{5} \)[/tex] cm
First, let's convert [tex]\( b_2 \)[/tex] from a mixed number to an improper fraction:
[tex]\[ b_2 = 12 \frac{1}{3} = 12 + \frac{1}{3} = 12 + 0.33333\ldots = 12.33333\ldots \][/tex]
The sum of the bases [tex]\( b_1 \)[/tex] and [tex]\( b_2 \)[/tex] is:
[tex]\[ b_1 + b_2 = 3.6 + 12.33333\ldots = 15.93333\ldots \][/tex]
Now, let's check the height [tex]\( \sqrt{5} \)[/tex]. The square root of 5 is an irrational number because it cannot be expressed as a fraction of two integers.
The values we need for the area calculation are:
- Sum of bases: [tex]\( b_1 + b_2 = 15.93333\ldots \)[/tex] cm (rational)
- Height: [tex]\( h = \sqrt{5} \)[/tex] cm (irrational)
According to the formula, the area [tex]\( A \)[/tex] is:
[tex]\[ A = \frac{1}{2} (b_1 + b_2) h \][/tex]
When an irrational number (the height, [tex]\( h = \sqrt{5} \)[/tex]) is multiplied by a rational number (the sum of the bases, [tex]\( b_1 + b_2 = 15.93333\ldots \)[/tex]), the result is generally irrational.
Thus, the area of the trapezoid is irrational because the height is irrational, and it is multiplied by the other rational dimensions.
