The radii of the base of two cylinders A and B are in the ratio 3:2 and their height in the ratio n:1. If the volume of cylinder A is 3 times that of cylinder B, the value of n is

Options

A. a.`(4)/(3)`
B. b.`(2)/(3)`
C. c.`(3)/(4)`
D. d.`(3)/(2)`

Answer

To solve the problem step by step, we will use the information provided about the two cylinders A and B.

### Step 1: Understand the given ratios
- The ratio of the radii of the bases of cylinders A and B is given as \(3:2\).
- The ratio of their heights is given as \(n:1\).

### Step 2: Write the formulas for the volumes
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \(r\) is the radius and \(h\) is the height.

### Step 3: Express the volumes of cylinders A and B
Let:
- Radius of cylinder A, \(r_A = 3k\) (for some constant \(k\))
- Radius of cylinder B, \(r_B = 2k\)
- Height of cylinder A, \(h_A = n\)
- Height of cylinder B, \(h_B = 1\)

Now, we can express the volumes:
- Volume of cylinder A, \(V_A = \pi (3k)^2 (n) = \pi \cdot 9k^2 \cdot n\)
- Volume of cylinder B, \(V_B = \pi (2k)^2 (1) = \pi \cdot 4k^2 \cdot 1\)

### Step 4: Set up the equation based on the volume relationship
According to the problem, the volume of cylinder A is 3 times that of cylinder B:
\[ V_A = 3 \times V_B \]
Substituting the expressions for \(V_A\) and \(V_B\):
\[ \pi \cdot 9k^2 \cdot n = 3 \cdot (\pi \cdot 4k^2) \]

### Step 5: Simplify the equation
We can cancel \(\pi\) and \(k^2\) from both sides (assuming \(k \neq 0\)):
\[ 9n = 3 \cdot 4 \]
\[ 9n = 12 \]

### Step 6: Solve for \(n\)
Now, divide both sides by 9:
\[ n = \frac{12}{9} = \frac{4}{3} \]

### Conclusion
The value of \(n\) is \(\frac{4}{3}\).

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