The ratio of weights of two sphere of different materials is 8:17 and the ratio of weights per 1 cc of materials of each is 289 : 64. The ratio of radii of the two spheres is

Options

A. `8:17`
B. `4:17`
C. `17:4`
D. `17:8`

Answer

To find the ratio of the radii of the two spheres, we can follow these steps:

### Step 1: Understand the Given Ratios
We are given:
- The ratio of weights of the two spheres: \( W_1 : W_2 = 8 : 17 \)
- The ratio of weights per cubic centimeter of the materials: \( D_1 : D_2 = 289 : 64 \)

### Step 2: Express the Ratios Mathematically
Let:
- \( W_1 = 8x \) and \( W_2 = 17x \) (for some \( x \))
- \( D_1 = 289y \) and \( D_2 = 64y \) (for some \( y \))

### Step 3: Relate Weight, Density, and Volume
The weight of an object can be expressed as:
\[ W = D \times V \]
Where \( V \) is the volume. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can write:
\[ W_1 = D_1 \times V_1 \]
\[ W_2 = D_2 \times V_2 \]

### Step 4: Express Volumes in Terms of Radii
Using the formula for volume:
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
\[ V_2 = \frac{4}{3} \pi r_2^3 \]
Substituting these into the weight equations:
\[ 8x = 289y \times \frac{4}{3} \pi r_1^3 \]
\[ 17x = 64y \times \frac{4}{3} \pi r_2^3 \]

### Step 5: Set Up the Ratio of Volumes
From the above equations, we can express the ratio of the weights:
\[ \frac{W_1}{W_2} = \frac{D_1 \times V_1}{D_2 \times V_2} \]
This gives us:
\[ \frac{8x}{17x} = \frac{289y \times \frac{4}{3} \pi r_1^3}{64y \times \frac{4}{3} \pi r_2^3} \]

### Step 6: Simplify the Equation
The \( \frac{4}{3} \pi \) and \( y \) terms cancel out:
\[ \frac{8}{17} = \frac{289 r_1^3}{64 r_2^3} \]

### Step 7: Rearranging the Equation
Cross-multiplying gives:
\[ 8 \times 64 r_2^3 = 17 \times 289 r_1^3 \]
\[ 512 r_2^3 = 4913 r_1^3 \]

### Step 8: Find the Ratio of Radii
Taking the ratio of the radii:
\[ \frac{r_1^3}{r_2^3} = \frac{512}{4913} \]
Taking the cube root:
\[ \frac{r_1}{r_2} = \sqrt[3]{\frac{512}{4913}} \]

### Step 9: Simplifying the Ratio
We can express \( 512 \) as \( 8^3 \) and \( 4913 \) as \( 17^3 \):
\[ \frac{r_1}{r_2} = \frac{8}{17} \]

### Conclusion
Thus, the ratio of the radii of the two spheres is:
\[ r_1 : r_2 = 8 : 17 \]