Question

find the volume of each solid. round to the nearest tenth. (example 3)
7.
8.

Explanation:

Step1: Calculate the volume of the first solid (two cones)

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). For the first solid, each cone has \(r = 8\space cm\). The total height of the two - cone solid is \(h_{total}=16 + 8=24\space cm\).
The volume of the two - cone solid \(V_{1}=2\times\frac{1}{3}\pi r^{2}h_{1}\) (where \(h_{1}\) is the height of one cone). Using \(V=\frac{1}{3}\pi r^{2}h_{total}\) (since \(V_{1}=\frac{1}{3}\pi r^{2}(h_{1}+h_{2})\) and \(h_{1}+h_{2}=h_{total}\)), substituting \(r = 8\space cm\) and \(h_{total}=24\space cm\):

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Step2: Calculate the volume of the second solid (cone + cylinder)

The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h_{cone}\) and for a cylinder is \(V_{cylinder}=\pi r^{2}h_{cylinder}\).
For the cone, \(r = 6\space cm\) and \(h_{cone}=10\space cm\). For the cylinder, \(r = 6\space cm\) and \(h_{cylinder}=8.5\space cm\).
The volume of the cone \(V_{cone}=\frac{1}{3}\pi\times6^{2}\times10=\frac{1}{3}\pi\times36\times 10 = 120\pi\)
The volume of the cylinder \(V_{cylinder}=\pi\times6^{2}\times8.5=\pi\times36\times8.5 = 306\pi\)
The total volume \(V_{2}=V_{cone}+V_{cylinder}=\pi(120 + 306)=426\pi\)

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Answer:

1. \(1607.7\space cm^{3}\)
2. \(1337.6\space cm^{3}\)