The ratio of the escape speed from two planets is 3 : 4 and the ratio of their masses is 9 : 16. What is the ratio of their radii ?

Explanation

To solve the problem, we need to find the ratio of the radii of two planets given the ratios of their escape speeds and masses.

### Step-by-Step Solution:

1. **Understanding Escape Speed**:
The escape speed (v) from a planet is given by the formula:
\[ v = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Setting Up Ratios**:
We are given:
- The ratio of escape speeds \( \frac{V_1}{V_2} = \frac{3}{4} \)
- The ratio of masses \( \frac{M_1}{M_2} = \frac{9}{16} \)

3. **Expressing Escape Speed Ratios**:
From the escape speed formula, we can express the ratio of escape speeds as:
\[ \frac{V_1}{V_2} = \sqrt{\frac{2GM_1}{R_1}} \div \sqrt{\frac{2GM_2}{R_2}} = \sqrt{\frac{M_1}{R_1}} \div \sqrt{\frac{M_2}{R_2}} \]
This simplifies to:
\[ \frac{V_1}{V_2} = \sqrt{\frac{M_1 R_2}{M_2 R_1}} \]

4. **Substituting Known Ratios**:
Now substituting the known ratios into the equation:
\[ \frac{3}{4} = \sqrt{\frac{M_1 R_2}{M_2 R_1}} \]
Substituting \( \frac{M_1}{M_2} = \frac{9}{16} \):
\[ \frac{3}{4} = \sqrt{\frac{9 R_2}{16 R_1}} \]

5. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ \left(\frac{3}{4}\right)^2 = \frac{9 R_2}{16 R_1} \]
This gives:
\[ \frac{9}{16} = \frac{9 R_2}{16 R_1} \]

6. **Cross Multiplying**:
Cross multiplying gives:
\[ 9 R_1 = 9 R_2 \]

7. **Simplifying the Equation**:
Dividing both sides by 9:
\[ R_1 = R_2 \]

8. **Finding the Ratio of Radii**:
Hence, the ratio of the radii \( \frac{R_1}{R_2} = 1 \) or:
\[ R_1 : R_2 = 1 : 1 \]

### Final Answer:
The ratio of the radii of the two planets is \( 1 : 1 \).