Given the three orthographic views of a solid as shown, find the volume of the solid.
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Step-by-Step Solution
Step 1: Analyze the orthographic views
The front view shows a trapezoid, indicating that the solid has varying depths or heights. The side view is a square with a diagonal, suggesting a prism-like structure or a wedge. The top view is an equilateral triangle with side length 2, which means the base of the solid is an equilateral triangle with side length 2.
Step 2: Determine the dimensions of the solid
From the top view, we know the base is an equilateral triangle with side length a=2. From the front view, the maximum height of the solid is h=2. The front view also shows that the solid can be decomposed into a triangular prism and a triangular pyramid.
Step 3: Calculate the area of the equilateral triangular base
The area of an equilateral triangle with side length a is given by the formula A=43a2. Substituting a=2, we find the area of the base to be 3 square units.
Step 4: Decompose the solid into simpler shapes
Looking at the front view, the solid can be decomposed into two parts: a triangular prism with a height of 1 unit and a triangular pyramid with a height of 1 unit. Both share the same equilateral triangular base.
Step 5: Calculate the volume of the triangular prism
The volume of a prism is calculated by multiplying its base area by its height. Here, the base area is 3 and the height of this part of the solid is 1, so the volume of the prism part is 3 cubic units.
Step 6: Calculate the volume of the triangular pyramid
The volume of a pyramid is one-third of the product of its base area and its height. For the pyramid part of the solid, the base area is 3 and the height is 1, resulting in a volume of 33 cubic units.
Step 7: Calculate the total volume of the solid
The total volume of the solid is the sum of the volumes of the triangular prism and the triangular pyramid. Adding 3 and 33 gives a total volume of 343 cubic units.