Example Question - cylinder volume
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Calculating the Circumference of a Cylinder's Base
<p>The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \) where \( r \) is the radius and \( h \) is the height. </p> <p>Given:</p> <p>\( V = 12,936 \text{ cm}^3 \)</p> <p>\( h = 21 \text{ cm} \)</p> <p>\( \pi \approx \frac{22}{7} \)</p> <p>To find the circumference \( C \), we need to find the radius \( r \) first.</p> <p>From the volume formula:</p> <p>\( 12,936 = \frac{22}{7} \times r^2 \times 21 \)</p> <p>Solving for \( r^2 \):</p> <p>\( r^2 = \frac{12,936 \times 7}{22 \times 21} \)</p> <p>\( r^2 = \frac{12,936}{66} \)</p> <p>\( r^2 = 196 \)</p> <p>\( r = \sqrt{196} \)</p> <p>\( r = 14 \text{ cm} \)</p> <p>Now we calculate the circumference \( C \) using the radius \( r \):</p> <p>\( C = 2 \pi r \)</p> <p>\( C = 2 \times \frac{22}{7} \times 14 \)</p> <p>\( C = 88 \text{ cm} \)</p> <p>Therefore, the circumference of the base of the cylinder is \( 88 \text{ cm} \).</p>
Calculation of Cylinder Dimensions Using Given Volume
Let the height of the cylinder be \( h \) and the radius be \( r \). The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \). The circumference \( C \) of the base of the cylinder is given by \( C = 2\pi r \). We are given: \( V = 12,936 \) cm\(^3\) \( \pi = \frac{22}{7} \) Firstly, we express \( r \) in terms of \( C \): <p>\( C = 2\pi r \Rightarrow r = \frac{C}{2\pi} \)</p> Substitute \( r \) into the volume formula and solve for \( h \): <p>\( V = \pi \left(\frac{C}{2\pi}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{2 \cdot \frac{22}{7}}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{7C}{44}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^{\!2} h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{4}\right)^{\!2} h \)</p> <p>\( 12,936 = \frac{22}{7} \cdot \frac{C^2}{16} \cdot h \)</p> <p>\( 12,936 \cdot \frac{16}{22} = C^2 h \)</p> <p>\( 12,936 \cdot \frac{16}{22} = C^2 h \)</p> <p>\( 12,936 \cdot \frac{7 \cdot 16}{22} = 7C^2 h \)</p> <p>\( 12,936 \cdot \frac{112}{22} = 7C^2 h \)</p> Now, let's solve for \( h \) in terms of \( C \): <p>\( h = \frac{12,936 \cdot \frac{112}{22}}{7C^2} \)</p> <p>\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)</p> Finally, we substitute the given value of the volume to find \( h \): <p>\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)</p>
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