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 Question:
This problem involves the geometric property of cylinders.
We are given that two cylinders, $A$ and $B$, have equal capacities (volumes).
We are also given the ratio of their diameters and need to find the ratio of their heights.
Using the cylinder volume formula, we can relate the radii and heights to find the required ratio.

Step 2: Key Formula or Approach:

• The volume (capacity) of a cylinder is given by the formula: \( V = \pi r^2 h \), where $r$ is the radius and $h$ is the height.
  
• Since the diameter ratio is equal to the radius ratio, we have: \( \frac{r_A}{r_B} = \frac{d_A}{d_B} = \frac{1}{4} \).
  
• Since the volumes are equal (\( V_A = V_B \)), we can write: \( \pi r_A^2 h_A = \pi r_B^2 h_B \).
  
• This simplifies to the ratio of heights: \( \frac{h_A}{h_B} = \left(\frac{r_B}{r_A}\right)^2 \).
  

Step 3: Detailed Explanation:

• Let the radii of cylinders $A$ and $B$ be $r_A$ and $r_B$, respectively.
  
• Let the heights of cylinders $A$ and $B$ be $h_A$ and $h_B$, respectively.
  
• The ratio of the diameters of cylinder $A$ to cylinder $B$ is given as $1 : 4$.
  
• Since radius is half of the diameter, the ratio of the radii is also $1 : 4$:
  \[ \frac{r_A}{r_B} = \frac{1}{4} \]
  
• This implies that:
  \[ r_B = 4 r_A \]
  
• The capacities of both cylinders are equal, meaning their volumes are identical:
  \[ V_A = V_B \]
  
• Substitute the volume formula for both cylinders into this relation:
  \[ \pi r_A^2 h_A = \pi r_B^2 h_B \]
  
• Divide both sides by $\pi$ to simplify:
  \[ r_A^2 h_A = r_B^2 h_B \]
  
• Rearrange the terms to express the ratio of heights $\frac{h_A}{h_B}$:
  \[ \frac{h_A}{h_B} = \frac{r_B^2}{r_A^2} = \left(\frac{r_B}{r_A}\right)^2 \]
  
• Substitute the ratio $\frac{r_B}{r_A} = \frac{4}{1}$ into the equation:
  \[ \frac{h_A}{h_B} = \left(\frac{4}{1}\right)^2 = \frac{16}{1} \]
  
• Thus, the ratio of the heights of cylinder $A$ to cylinder $B$ is $16 : 1$.
  

Step 4: Final Answer:
The ratio of the heights of cylinders A and B is $16 : 1$, which matches Option (B).