Question

half of a cylinder is attached to a triangular prism as shown in the diagram below. the diameter of the half cylinder is 8 inches, and the height of the cylinder is 12 inches. the triangular faces have bases of 8 inches and perpendicular heights of 10 inches. what is the volume, v, of this composite solid? leave your answer in terms of π. 4 of 4 questions v = 960 + 192 π in³ v = 960 + 96 π in³ v = 480 + 96 π in³ v = 480 + 192 π in³

Explanation:

Step1: Calculate the volume of the triangular prism

The formula for the volume of a triangular prism is \(V_{prism}=B\times h\), where \(B\) is the area of the triangular base and \(h\) is the height of the prism.
The area of a triangle \(B = \frac{1}{2}\times base\times height\). Given \(base = 8\) inches and \(height = 10\) inches, \(B=\frac{1}{2}\times8\times10=40\) square inches.
The height of the prism \(h = 12\) inches. So \(V_{prism}=40\times12 = 480\) cubic inches.

Step2: Calculate the volume of the half - cylinder

The formula for the volume of a full - cylinder is \(V_{cylinder}=\pi r^{2}h\). Given the diameter \(d = 8\) inches, the radius \(r=\frac{d}{2}=4\) inches and the height \(h = 12\) inches.
The volume of the half - cylinder \(V_{half - cylinder}=\frac{1}{2}\pi r^{2}h\). Substitute \(r = 4\) and \(h = 12\) into the formula: \(V_{half - cylinder}=\frac{1}{2}\pi\times4^{2}\times12\).
First, calculate \(4^{2}=16\), then \(\frac{1}{2}\times16\times12 = 96\). So \(V_{half - cylinder}=96\pi\) cubic inches.

Step3: Calculate the volume of the composite solid

The volume of the composite solid \(V=V_{prism}+V_{half - cylinder}\).
Substitute \(V_{prism}=480\) and \(V_{half - cylinder}=96\pi\) into the formula: \(V = 480+96\pi\) cubic inches.

Answer:

\(V = 480 + 96\pi\mathrm{in}^{3}\) (the third option)