The ratio of the weights of two spheres is 8:27 and the ratio of weights per 1 cc of materials of two is 8:1. The ratio of the radii of the spheres is

Options

A. `2:3`
B. `1:3`
C. `3:1`
D. `3:2`

Answer

To solve the problem, we need to find the ratio of the radii of two spheres given the ratio of their weights and the ratio of their weights per cubic centimeter of material.

### Step-by-Step Solution:

1. **Understanding the Given Ratios:**
- The ratio of the weights of the two spheres (A and B) is given as:
\[ \text{Weight of A} : \text{Weight of B} = 8 : 27 \]
- The ratio of weights per cubic centimeter (density) of the two spheres is:
\[ \text{Density of A} : \text{Density of B} = 8 : 1 \]

2. **Using the Formula for Mass:**
- The mass (weight) of a sphere can be expressed as:
\[ \text{Mass} = \text{Volume} \times \text{Density} \]
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Therefore, the mass of sphere A can be written as:
\[ \text{Mass of A} = \frac{4}{3} \pi R^3 \times d_1 \]
- And for sphere B:
\[ \text{Mass of B} = \frac{4}{3} \pi r^3 \times d_2 \]

3. **Setting Up the Ratio of Masses:**
- From the problem, we know:
\[ \frac{\text{Mass of A}}{\text{Mass of B}} = \frac{8}{27} \]
- Substituting the expressions for mass:
\[ \frac{\frac{4}{3} \pi R^3 \times d_1}{\frac{4}{3} \pi r^3 \times d_2} = \frac{8}{27} \]
- The \( \frac{4}{3} \pi \) cancels out:
\[ \frac{R^3 \times d_1}{r^3 \times d_2} = \frac{8}{27} \]

4. **Substituting the Density Ratio:**
- We know the density ratio \( \frac{d_1}{d_2} = \frac{8}{1} \):
\[ d_1 = 8d_2 \]
- Substituting this into the mass ratio:
\[ \frac{R^3 \times 8d_2}{r^3 \times d_2} = \frac{8}{27} \]
- The \( d_2 \) cancels out:
\[ \frac{8R^3}{r^3} = \frac{8}{27} \]

5. **Simplifying the Equation:**
- Dividing both sides by 8:
\[ \frac{R^3}{r^3} = \frac{1}{27} \]

6. **Finding the Ratio of Radii:**
- Taking the cube root of both sides:
\[ \frac{R}{r} = \frac{1}{3} \]
- Therefore, the ratio of the radii of the spheres is:
\[ R : r = 1 : 3 \]

### Final Answer:
The ratio of the radii of the spheres is \( 1 : 3 \).

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