Question

what is the volume of the figure below if a = 9 units and b = 7 units?
note: figure is not drawn to scale.
a. 760.5 cubic units
b. 1,012.5 cubic units
c. 1,296 cubic units
d. 364.5 cubic units

Explanation:

Step1: Calculate the volume of the cube

The volume of a cube with side length \(a\) is \(V_{cube}=a^{3}\). Given \(a = 9\) units, so \(V_{cube}=9^{3}=729\) cubic units.

Step2: Calculate the volume of the pyramid

The volume of a pyramid is \(V_{pyramid}=\frac{1}{3}×\text{base area}×\text{height}\). The base of the pyramid is a square with side - length \(a = 9\) units (same as the side of the cube), so the base area \(A=a^{2}=81\) square units and the height \(h = b=7\) units. Then \(V_{pyramid}=\frac{1}{3}×81×7 = 189\) cubic units.

Step3: Calculate the total volume

The total volume \(V = V_{cube}+V_{pyramid}\). Substitute the values: \(V=729 + 189=918\) (This is wrong, let's re - check. Wait, no, the figure is a combination of a cube and a triangular - based pyramid? No, looking at the formula again. Wait, the formula for the volume of the combined figure: assume it's a cube (\(V_{1}=a^{3}\)) and a pyramid. Wait, no, if we consider the correct formula. The figure is a cube (volume \(V_{cube}=a^{3}\)) and a pyramid. The base of the pyramid is a square (area \(A = a^{2}\)) and height \(b\). The volume of the pyramid \(V_{pyramid}=\frac{1}{2}×a^{2}×b\) (if it's a triangular - based pyramid formed by splitting a square - based pyramid). Wait, no, another approach:
The figure is composed of a cube (\(a\times a\times a\)) and a triangular prism - like part (but no, using the formula for the volume of the combined solid.
The correct formula: The volume of the cube \(V_{cube}=a^{3}\) and the volume of the pyramid (assuming it's a pyramid with base area \(a^{2}\) and height \(b\)). The volume of the pyramid \(V_{pyramid}=\frac{1}{2}a^{2}b\) (derived from the fact that the cross - section is a triangle).
\(V=a^{3}+\frac{1}{2}a^{2}b\)
Substitute \(a = 9\) and \(b = 7\)
\(V=9^{3}+\frac{1}{2}×9^{2}×7\)
\(V = 729+\frac{1}{2}×81×7\)
\(V=729 + 283.5\)
\(V = 1012.5\)

Answer:

B. \(1,012.5\) cubic units