Question

If the area of the cross-section of a cylinder is doubled, what will be the ratio of its height and radius of the base?

• A. 1:2
• B. 2:1
• C. 1:1
• D. 3:1

Hint:

When the cross-sectional area of a cylinder is doubled, the radius increases by \( \sqrt{2} \), and the height must also change accordingly to maintain the volume.

Answer 1

The Correct Option is B

Let the radius of the base be \( r \), and the height of the cylinder be \( h \). The area of the cross-section (which is the area of the circular base) is given by: \[ A = \pi r^2 \] The volume \( V \) of the cylinder is given by: \[ V = A \times h = \pi r^2 \times h \] We are told that the area of the cross-section is doubled, meaning the new area \( A' = 2A = 2\pi r^2 \). Since the area of the base is proportional to the square of the radius, if the area is doubled, the radius will increase by a factor of \( \sqrt{2} \). Now, let's find the ratio of the new height \( h' \) to the new radius \( r' \). Since the volume of the cylinder remains constant, the volume equation becomes: \[ V = A' \times h' = \pi r'^2 \times h' \] Using the relationship \( A' = 2A \), we substitute the values: \[ \pi r^2 \times h = \pi r'^2 \times h' \] Substituting \( r' = \sqrt{2}r \), we get: \[ \pi r^2 \times h = \pi (2r^2) \times h' \] Simplifying: \[ h = 2h' \] Thus, the ratio of height and radius is \( 2:1 \).