Question - Calculation of Cylinder Dimensions Using Given Volume
Solution:
\( C = 2\pi r \Rightarrow r = \frac{C}{2\pi} \)
Substitute \( r \) into the volume formula and solve for \( h \):\( V = \pi \left(\frac{C}{2\pi}\right)^2 h \)
\( 12,936 = \frac{22}{7} \left(\frac{C}{2 \cdot \frac{22}{7}}\right)^2 h \)
\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^2 h \)
\( 12,936 = \frac{22}{7} \left(\frac{7C}{44}\right)^2 h \)
\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^{\!2} h \)
\( 12,936 = \frac{22}{7} \left(\frac{C}{4}\right)^{\!2} h \)
\( 12,936 = \frac{22}{7} \cdot \frac{C^2}{16} \cdot h \)
\( 12,936 \cdot \frac{16}{22} = C^2 h \)
\( 12,936 \cdot \frac{16}{22} = C^2 h \)
\( 12,936 \cdot \frac{7 \cdot 16}{22} = 7C^2 h \)
\( 12,936 \cdot \frac{112}{22} = 7C^2 h \)
Now, let's solve for \( h \) in terms of \( C \):\( h = \frac{12,936 \cdot \frac{112}{22}}{7C^2} \)
\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)
Finally, we substitute the given value of the volume to find \( h \):\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)
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