Question:
Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.
\includegraphics[width=0.4\linewidth]{q7 MA.PNG}
The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is:
Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.
\includegraphics[width=0.4\linewidth]{q7 MA.PNG}
The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is:
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For geometry problems, calculate volumes or areas separately for each component and ensure proportions match the given context.
Updated On: Feb 3, 2026
- \( 3/2 \)
- \( 7\pi/2 \)
- \( 9/2 \)
- \( 3 \)
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The Correct Option is B
Solution and Explanation
Step 1: Determining the occupied space.
The occupied space consists of seven identical cylindrical chalk-sticks. The volume of one chalk-stick is given by:
\[
V_{{chalk}} = \pi r^2 h,
\]
where \( r \) is the radius and \( h \) is the height (length). For seven chalk-sticks:
\[
V_{{occupied}} = 7 \pi r^2 h.
\]
Step 2: Determining the total space.
The volume of the cylindrical container is:
\[
V_{{container}} = \pi R^2 h,
\]
where \( R \) is the radius of the container. Given the tight fit of the chalk-sticks, the area of the base of the container is proportional to the arrangement of seven cylinders.
Step 3: Ratio calculation.
The ratio of occupied to empty space is:
\[
\frac{V_{{occupied}}}{V_{{empty}}} = \frac{7 \pi r^2 h}{\pi R^2 h - 7 \pi r^2 h} = \frac{7\pi/2}.
\]
Step 4: Conclusion.
The ratio is \( {(2) } 7\pi/2 \).
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