Answer:
294 ft³
Step-by-step explanation:
To find the volume of a triangular prism, we must multiply the triangle base by the height. In this problem, we must first find the height of the
10 ft x 7ft x 7ft triangle to find the area of the triangle. From there, we can multiply that area by the height of 12 ft to get the volume.
To find the height of the triangle, draw a straight line from the intersection of the two 7 ft sides all the way down to the 10 ft side to create two equal right triangles with side lengths 7 ft and 5 ft. Since the 7 ft side is opposite to the right angle created it is the hypotenuse. From there, use the pythagorean theorem to find the height and then the area of the triangle.
Solving:
[tex]\text{Pythagorean Therom:} ~~a^2 + b^2 = c^2 ~ (a = 5, c =7, b =?)\\\text{Height(b)} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} = \sqrt{24}~\boxed{ \approx 4.9 \text{ ft}}[/tex]
Now find the area of the triangle:
[tex]\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 4.9 = \boxed{24.5 \text{ square feet}}\][/tex]
Now multiply by the height to get the volume:
[tex]\[\text{Volume} = \text{Base area} \times \text{length} = 24.5 \times 12 = \boxed{294 \text{ cubic feet}}\][/tex]
Therefore, the volume is approximately 294 ft³.
