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Volume of Composite Cylinders Theory
![\begin{multicols}{2} \textbf{Example 1}\\ Find the exact volume of the composite solid.\\ \begin{center} \begin{tikzpicture}[scale=0.3,line width=1pt] \begin{scope}[shift=(left:6.2)] \coordinate[label=left:] (l) at (-90:10); \end{scope} \draw (left:6.2) edge (l); \begin{scope}[shift=(right:6.2)] \coordinate[label=left:] (r) at (-90:10); \end{scope} \draw (right:6.2) edge (r) node[right,yshift=-2cm] {\scriptsize 10 cm}; \draw[out=240,in=-60,looseness=0.5] (r) to (l); \draw[out=-240,in=60,looseness=0.5,dashed] (r) to (l); \draw (0,0) circle (2pt); \draw[] (0,0)--(60:3.2) node[right,pos=0.7] {\scriptsize 6 cm}; \draw[] (0,0)--(left:4) node[below=-1.5pt,pos=0.5] {\scriptsize 4 cm}; \draw (0,0) circle [x radius=6.2cm, y radius=2.8cm]; \draw (0,0) circle [x radius=4cm, y radius=1.5cm]; \end{tikzpicture} \end{center} \textbf{Example 1 solution}\\ \(\begin{aligned} \text { Volume of small cylinder } & =\pi \times 4^2 \times 10 \\ & =160 \pi \text{ cm}^3 \\ \text { Volume of large cylinder } & =\pi \times 6^2 \times 10 \\ & =360 \pi \text{ cm}^3 \end{aligned}\)\\ \(\begin{aligned} \therefore \text { Volume of soled } & =360 \pi-160 \pi \\ & =200 \pi \text{ cm}^3 \end{aligned}\) \columnbreak \textbf{Example 2}\\ Find the exact volume of the composite solid.\\ \begin{center} \begin{tikzpicture}[scale=0.4,line width=1pt] \begin{scope}[shift=(left:6.2)] \coordinate[label=left:] (l) at (-90:4); \end{scope} \draw (left:6.2) edge (l); \begin{scope}[shift=(right:6.2)] \coordinate[label=left:] (r) at (-90:4); \end{scope} \draw (right:6.2) edge (r); \draw[out=240,in=-60,looseness=0.5] (r) to (l); \draw (0,0) circle [x radius=6.2cm, y radius=2cm]; %small cylinder \coordinate[label=left:] (u) at (90:4); \begin{scope}[shift=(u)] \coordinate[label=left:] (v) at (left:3); \end{scope} \begin{scope}[shift=(u)] \coordinate[label=left:] (w) at (right:3); \end{scope} \begin{scope}[shift=(v)] \coordinate[label=left:] (l1) at (-90:4); \end{scope} \draw (v) edge (l1); \begin{scope}[shift=(w)] \coordinate[label=left:] (r1) at (-90:4); \end{scope} \draw (w) edge (r1); \draw[out=260,in=-80,looseness=0.5] (l1) to (r1); \draw (u) circle [x radius=3cm, y radius=1cm]; \path (r)--(right:6.2) node[right,midway] {4cm}; \path (w)--(r1) node[right,pos=0.4] {4cm}; \draw (u)--(w) node[above=10pt,pos=0.35] {3cm}; \draw[fill=black] (u) circle (3pt); \begin{scope}[shift=(-90:6)] \draw[|-|] (left:6.2)--(right:6.2) node[below,pos=0.5] {8cm}; \end{scope} \end{tikzpicture} \end{center} \textbf{Example 2 solution}\\ \(\begin{aligned} \text { Volume of small cylinder } & =\pi \times 3^2 \times 4 \\ & =36 \pi \text{ cm}^3 \end{aligned}\)\\ \(\begin{aligned} \text { Volume of large cylinder } & =\pi \times 4^2 \times 4 \\ & =64 \pi \text{ cm}^3 \end{aligned}\)\\ \(\begin{aligned} \therefore \text { Volume of solid } & =36 \pi+64 \pi \\ & =100 \pi \text{ cm}^3 \end{aligned}\) \end{multicols}](/media/huxfxqa1/7995.png)
NSW Y8 Maths Volume Volume of Composite Cylinders
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