Question

Let A and B be two cylinders such that the capacity of A is the same as the capacity of B. The ratio of the diameters of A and B is 1 : 4. What is the ratio of the heights of A and B?

• A. 16 : 3
• B. 16 : 1
• C. 1 : 16
• D. 3 : 16

Hint:

Since volume \( V \propto r^2 h \), for constant volume, height is inversely proportional to the square of the radius (\( h \propto \frac{1}{r^2} \)).
The radius ratio is $1 : 4$.
Squaring this ratio gives $1 : 16$.
Taking the inverse of this ratio gives the height ratio: $16 : 1$.
This mental calculation avoids writing equations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Capacity refers to the volume of the cylinders.
The volume of a cylinder depends on its radius and height.
Since the volumes are equal, any change in the radius must be balanced by an inverse change in the square of the height.
Diameter ratio is the same as the radius ratio.
Step 2: Key Formula or Approach:
1. Volume \( V = \pi r^2 h \).
2. If \( V_A = V_B \), then \( \pi r_A^2 h_A = \pi r_B^2 h_B \).
3. This implies \( r_A^2 h_A = r_B^2 h_B \).
Step 2: Detailed Explanation:
Let the radii be \( r_A \) and \( r_B \), and heights be \( h_A \) and \( h_B \).
Ratio of diameters \( d_A : d_B = 1 : 4 \), which implies \( r_A : r_B = 1 : 4 \).
Let \( r_A = x \) and \( r_B = 4x \).
Since volumes are equal:
\[ \pi (x)^2 h_A = \pi (4x)^2 h_B \]
Cancel \( \pi \) and simplify the squares:
\[ x^2 h_A = 16x^2 h_B \]
Divide both sides by \( x^2 \):
\[ h_A = 16 h_B \]
To find the ratio \( h_A : h_B \), we divide by \( h_B \):
\[ \frac{h_A}{h_B} = \frac{16}{1} \]
The ratio of heights is 16 : 1.
Step 3: Final Answer:
The ratio of the heights is 16 : 1.
This matches Option (B).