In two spheres radius of one is half of the other, then volume of second sphere with respect to first, is

Options

A. double
B. four times
C. eight times
D. `(22)/(7)` times

Answer

To find the volume of the second sphere with respect to the first sphere, we start by using the formula for the volume of a sphere.

### Step 1: Write the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Define the radii of the two spheres.
Let the radius of the first sphere (Sphere 1) be \( r \). According to the problem, the radius of the second sphere (Sphere 2) is half of the first sphere's radius. Therefore, we have:
\[ \text{Radius of Sphere 1} (r_1) = r \]
\[ \text{Radius of Sphere 2} (r_2) = \frac{r}{2} \]

### Step 3: Calculate the volume of Sphere 1.
Using the volume formula for Sphere 1:
\[ V_1 = \frac{4}{3} \pi (r_1)^3 = \frac{4}{3} \pi r^3 \]

### Step 4: Calculate the volume of Sphere 2.
Using the volume formula for Sphere 2:
\[ V_2 = \frac{4}{3} \pi (r_2)^3 = \frac{4}{3} \pi \left(\frac{r}{2}\right)^3 \]
Calculating \( \left(\frac{r}{2}\right)^3 \):
\[ \left(\frac{r}{2}\right)^3 = \frac{r^3}{8} \]
Thus, the volume of Sphere 2 becomes:
\[ V_2 = \frac{4}{3} \pi \left(\frac{r^3}{8}\right) = \frac{4}{3} \cdot \frac{\pi r^3}{8} = \frac{4 \pi r^3}{24} = \frac{\pi r^3}{6} \]

### Step 5: Find the ratio of the volumes of Sphere 2 to Sphere 1.
Now, we need to find the ratio of the volume of Sphere 2 to the volume of Sphere 1:
\[ \frac{V_2}{V_1} = \frac{\frac{\pi r^3}{6}}{\frac{4}{3} \pi r^3} \]
Cancelling \( \pi r^3 \) from the numerator and denominator:
\[ \frac{V_2}{V_1} = \frac{1/6}{4/3} = \frac{1}{6} \cdot \frac{3}{4} = \frac{3}{24} = \frac{1}{8} \]

### Conclusion:
The volume of the second sphere with respect to the first sphere is:
\[ \frac{V_2}{V_1} = \frac{1}{8} \]