By melting two solid metallic spheres of radii 1 cm and 6 cm a hollow sphere of thickness 1 cm is made. The external radius of the hollow sphere will be

Options

A. 9 cm
B. 6 cm
C. 7 cm
D. 8 cm

Answer

To solve the problem step by step, we need to find the external radius of the hollow sphere formed by melting two solid metallic spheres.

### Step 1: Calculate the volumes of the two solid spheres
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For the first sphere with radius \( r_1 = 1 \) cm:
\[ V_1 = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \text{ cm}^3 \]

For the second sphere with radius \( r_2 = 6 \) cm:
\[ V_2 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

### Step 2: Calculate the total volume of the two spheres
Now, we add the volumes of both spheres:
\[ V_{total} = V_1 + V_2 = \frac{4}{3} \pi + 288 \pi = \left(\frac{4}{3} + 288\right) \pi = \left(\frac{4 + 864}{3}\right) \pi = \frac{868}{3} \pi \text{ cm}^3 \]

### Step 3: Set up the equation for the hollow sphere
Let the internal radius of the hollow sphere be \( x \) cm. The external radius will then be \( x + 1 \) cm (since the thickness of the hollow sphere is 1 cm). The volume of the hollow sphere can be expressed as:
\[ V_{hollow} = \frac{4}{3} \pi \left((x + 1)^3 - x^3\right) \]

### Step 4: Equate the volumes
Setting the total volume equal to the volume of the hollow sphere:
\[ \frac{868}{3} \pi = \frac{4}{3} \pi \left((x + 1)^3 - x^3\right) \]

### Step 5: Simplify the equation
Cancel \( \frac{4}{3} \pi \) from both sides:
\[ 868 = 4 \left((x + 1)^3 - x^3\right) \]

Now, simplify the right-hand side:
\[ (x + 1)^3 - x^3 = (x^3 + 3x^2 + 3x + 1) - x^3 = 3x^2 + 3x + 1 \]
Thus, we have:
\[ 868 = 4(3x^2 + 3x + 1) \]
\[ 868 = 12x^2 + 12x + 4 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 12x^2 + 12x + 4 - 868 = 0 \]
\[ 12x^2 + 12x - 864 = 0 \]
Dividing the entire equation by 12:
\[ x^2 + x - 72 = 0 \]

### Step 7: Factor the quadratic equation
Factoring the quadratic:
\[ (x - 8)(x + 9) = 0 \]
Thus, \( x = 8 \) or \( x = -9 \). Since radius cannot be negative, we take \( x = 8 \) cm.

### Step 8: Find the external radius
The external radius \( R \) is:
\[ R = x + 1 = 8 + 1 = 9 \text{ cm} \]

### Final Answer
The external radius of the hollow sphere is **9 cm**.

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