The ratio of total surface area and volume of a sphere is 1 : 7. This sphere is melted to form small spheres of equal size. The radius of each small sphere is `1// 6^(th)` the radius of the large sphere. What is the sum (in `cm^(2)`) of curved surface areas of all small spheres?

Options

A. 31276
B. 36194
C. 25182
D. 33264

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values to find the sum of the curved surface areas of all small spheres.

### Step 1: Understand the Ratio of Surface Area and Volume
The ratio of the total surface area (TSA) to the volume (V) of a sphere is given as 1:7.

The formulas for the total surface area and volume of a sphere are:
- Total Surface Area (TSA) = \(4\pi r^2\)
- Volume (V) = \(\frac{4}{3}\pi r^3\)

Given the ratio:
\[ \frac{TSA}{V} = \frac{1}{7} \]

### Step 2: Set Up the Equation
Substituting the formulas into the ratio:
\[ \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{1}{7} \]

### Step 3: Simplify the Equation
Cancelling \(4\pi\) from both sides:
\[ \frac{r^2}{\frac{1}{3}r^3} = \frac{1}{7} \]
This simplifies to:
\[ \frac{3}{r} = \frac{1}{7} \]

### Step 4: Solve for r
Cross-multiplying gives:
\[ 3 \cdot 7 = r \implies r = 21 \text{ cm} \]

### Step 5: Determine the Radius of the Small Spheres
The radius of each small sphere is given as \(\frac{1}{6}\) of the radius of the large sphere:
\[ r' = \frac{1}{6} \times 21 = 3.5 \text{ cm} \]

### Step 6: Calculate the Volume of the Large Sphere
Using the volume formula:
\[ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (21)^3 \]

### Step 7: Calculate the Volume of Each Small Sphere
Using the radius of the small sphere:
\[ V' = \frac{4}{3}\pi (r')^3 = \frac{4}{3}\pi (3.5)^3 \]

### Step 8: Find the Number of Small Spheres
Since the large sphere is melted to form small spheres, we can find the number of small spheres (n):
\[ n = \frac{V}{V'} = \frac{\frac{4}{3}\pi (21)^3}{\frac{4}{3}\pi (3.5)^3} \]
The \(\frac{4}{3}\pi\) cancels out:
\[ n = \frac{(21)^3}{(3.5)^3} \]

### Step 9: Calculate n
Calculating \(n\):
\[ n = \frac{21^3}{(3.5)^3} = \frac{21^3}{\left(\frac{21}{6}\right)^3} = 6^3 = 216 \]

### Step 10: Calculate the Curved Surface Area of All Small Spheres
The curved surface area (CSA) of one small sphere is:
\[ CSA' = 4\pi (r')^2 = 4\pi (3.5)^2 \]
The total curved surface area of all small spheres is:
\[ Total CSA = n \times CSA' = 216 \times 4\pi (3.5)^2 \]

### Step 11: Final Calculation
Calculating the total CSA:
\[ Total CSA = 216 \times 4\pi \times 12.25 = 216 \times 4 \times \frac{22}{7} \times 12.25 \]
Calculating this gives:
\[ Total CSA = 33264 \text{ cm}^2 \]

### Final Answer
The sum of the curved surface areas of all small spheres is \(33264 \text{ cm}^2\).

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