To determine the volume of the given prism, we need to follow these steps.
1. Identify the shape of the base: The base of this prism is a 45-45-90 triangle. In such triangles, the two legs are equal in length.
2. Calculate the area of the base: The formula for the area of a triangle is [tex]\(\frac{1}{2} \times \text{base} \times \text{height}\)[/tex]. Since both legs are 5 inches, we use one leg as the base and the other leg as the height:
[tex]\[
\text{Area of the base} = \frac{1}{2} \times 5 \times 5 = \frac{1}{2} \times 25 = 12.5 \, \text{square inches}
\][/tex]
So, the area of the base is [tex]\(12.5 \, \text{square inches}\)[/tex].
3. Find the height of the prism: According to the problem, the height of the prism is [tex]\(1.8 \, \text{inches}\)[/tex].
4. Calculate the volume of the prism: The volume of a prism is determined by multiplying the area of the base by the height of the prism:
[tex]\[
\text{Volume} = \text{area of the base} \times \text{height} = 12.5 \, \text{square inches} \times 1.8 \, \text{inches} = 22.5 \, \text{cubic inches}
\][/tex]
Thus, the volume of the prism is [tex]\(22.5 \, \text{cubic inches}\)[/tex].
Therefore, the best answer from the choices provided is:
C. [tex]\(22.5 \, \text{in}^3\)[/tex]
