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:

• A. \( 3/2 \)
• B. \( 7\pi/2 \)
• C. \( 9/2 \)
• D. \( 3 \)

Hint:

For geometry problems, calculate volumes or areas separately for each component and ensure proportions match the given context.

Answer 1

The Correct Option is B

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 \( {(2) } 7\pi/2 \).