The ratio of the volumes of two cylinders is 7:3 and the ratio of their heights is 7:9. If the area of the base of the second cylinder is `154 cm^(2)`, then what will be the radius (in cm) of the first cylinder ?

Options

A. `6sqrt(2)`
B. `6sqrt(3)`
C. `7sqrt(2)`
D. `7sqrt(3)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given ratios
We know that the ratio of the volumes of two cylinders is 7:3, and the ratio of their heights is 7:9.

### Step 2: Write the formula for the volume of a cylinder
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: Set up the equations based on the ratios
Let the radius and height of the first cylinder be \( r_1 \) and \( h_1 \), and for the second cylinder, let them be \( r_2 \) and \( h_2 \).
From the ratios:
- Volume ratio:
\[ \frac{V_1}{V_2} = \frac{7}{3} \]
\[ \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \frac{7}{3} \]
This simplifies to:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{7}{3} \]

- Height ratio:
\[ \frac{h_1}{h_2} = \frac{7}{9} \]
Let \( h_1 = 7k \) and \( h_2 = 9k \) for some constant \( k \).

### Step 4: Substitute heights into the volume ratio equation
Substituting \( h_1 \) and \( h_2 \) into the volume ratio gives:
\[ \frac{r_1^2 (7k)}{r_2^2 (9k)} = \frac{7}{3} \]
The \( k \) cancels out:
\[ \frac{r_1^2}{r_2^2} \cdot \frac{7}{9} = \frac{7}{3} \]

### Step 5: Solve for the ratio of the radii
Cross-multiplying gives:
\[ 3r_1^2 \cdot 7 = 7r_2^2 \cdot 9 \]
This simplifies to:
\[ 3r_1^2 = 9r_2^2 \]
Dividing both sides by 3:
\[ r_1^2 = 3r_2^2 \]
Taking the square root:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{3}} \]

### Step 6: Express \( r_1 \) in terms of \( r_2 \)
Let \( r_2 = x \), then:
\[ r_1 = \frac{x}{\sqrt{3}} \]

### Step 7: Find the area of the base of the second cylinder
The area of the base of the second cylinder is given as \( 154 \, \text{cm}^2 \):
\[ \pi r_2^2 = 154 \]
Using \( \pi \approx 22/7 \):
\[ \frac{22}{7} r_2^2 = 154 \]
Multiplying both sides by \( 7 \):
\[ 22 r_2^2 = 1078 \]
Dividing by 22:
\[ r_2^2 = \frac{1078}{22} = 49 \]
Taking the square root:
\[ r_2 = 7 \, \text{cm} \]

### Step 8: Calculate \( r_1 \)
Now substituting \( r_2 \) back into the equation for \( r_1 \):
\[ r_1 = \frac{7}{\sqrt{3}} \]

### Final Answer
Thus, the radius of the first cylinder is:
\[ r_1 = \frac{7}{\sqrt{3}} \, \text{cm} \]