To determine the area of the triangular gable, we first need to convert the measurements of the sides into a consistent unit. The sides of the right triangle are given as [tex]\(8\)[/tex] feet and [tex]\(4\)[/tex] inches each. Let's start by converting these measurements to inches, and then to feet, so we can perform our calculations accurately.
1. Conversion to inches:
- Since there are 12 inches in a foot, first, we convert 8 feet to inches:
[tex]\[
8 \text{ feet} \times 12 \text{ inches per foot} = 96 \text{ inches}
\][/tex]
- Add the additional 4 inches:
[tex]\[
96 \text{ inches} + 4 \text{ inches} = 100 \text{ inches}
\][/tex]
Therefore, each side of the triangle is [tex]\(100\)[/tex] inches.
2. Conversion to feet:
- We convert the total inches back to feet:
[tex]\[
\frac{100 \text{ inches}}{12 \text{ inches per foot}} = 8.3333\ldots \text{ feet}
\][/tex]
We can represent this result more accurately as [tex]\(8.333\)[/tex] feet.
3. Calculate the area of the right triangle:
- The formula for the area of a right triangle is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
- In this case, both the base and the height are [tex]\(8.333\)[/tex] feet:
[tex]\[
\text{Area} = \frac{1}{2} \times 8.333 \text{ feet} \times 8.333 \text{ feet} = 34.7222\ldots \text{ square feet}
\][/tex]
- Rounding this to the nearest hundredth, we get:
[tex]\[
34.72 \text{ square feet}
\][/tex]
The total area the painter needs to paint is [tex]\(\boxed{34.72}\)[/tex] square feet.
