A large sphere of radius 'R' cm was converted into 64 small spheres of radius 'r' cm and then one small sphere is converted into 16 smaller cones of radius of 'a' cm. If height of cone is two times of its radius, then find R:a:r.

Options

A. A)`6:1:2`
B. B)`4:2:1 `
C. C)`8:1:2`
D. D)`4:1:2`

Answer

To solve the problem, we need to find the ratio \( R : a : r \) based on the given conditions. Let's break it down step by step.

### Step 1: Volume of the Large Sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the large sphere.

### Step 2: Volume of the Small Spheres
The large sphere is converted into 64 small spheres, each with radius \( r \). Therefore, the total volume of the small spheres is:
\[ 64 \times \frac{4}{3} \pi r^3 \]

### Step 3: Setting Volumes Equal
Since the volume of the large sphere is equal to the total volume of the small spheres, we have:
\[ \frac{4}{3} \pi R^3 = 64 \times \frac{4}{3} \pi r^3 \]
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 64 r^3 \]

### Step 4: Finding the Ratio \( R : r \)
Taking the cube root of both sides gives:
\[ R = 4r \]
Thus, we can express the ratio:
\[ \frac{R}{r} = 4 \quad \Rightarrow \quad R : r = 4 : 1 \]

### Step 5: Volume of One Small Sphere
Now, we consider one of the small spheres with radius \( r \). Its volume is:
\[ \frac{4}{3} \pi r^3 \]

### Step 6: Volume of the Cones
This small sphere is converted into 16 smaller cones, each with radius \( a \) and height \( h = 2a \). The volume \( V \) of one cone is given by:
\[ V = \frac{1}{3} \pi a^2 h = \frac{1}{3} \pi a^2 (2a) = \frac{2}{3} \pi a^3 \]
Thus, the total volume of the 16 cones is:
\[ 16 \times \frac{2}{3} \pi a^3 = \frac{32}{3} \pi a^3 \]

### Step 7: Setting Volumes Equal Again
Since the volume of the small sphere is equal to the total volume of the cones, we have:
\[ \frac{4}{3} \pi r^3 = \frac{32}{3} \pi a^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides gives:
\[ r^3 = 8a^3 \]

### Step 8: Finding the Ratio \( r : a \)
Taking the cube root of both sides gives:
\[ r = 2a \]
Thus, we can express the ratio:
\[ \frac{r}{a} = 2 \quad \Rightarrow \quad r : a = 2 : 1 \]

### Step 9: Final Ratio \( R : a : r \)
Now we have:
- \( R : r = 4 : 1 \)
- \( r : a = 2 : 1 \)

To find \( R : a : r \), we can express \( r \) in terms of \( a \):
\[ r = 2a \quad \Rightarrow \quad R = 4r = 4(2a) = 8a \]

Thus, the ratios can be written as:
\[ R : a : r = 8a : a : 2a = 8 : 1 : 2 \]

### Final Answer
The required ratio is:
\[ R : a : r = 8 : 1 : 2 \]