A solid metalic sphere, 3 cm in radius is melted and recast into three spherical balls with radii 1.5 cm, 2 cm and x cm. Find the value of x.

Answer

To solve the problem, we need to find the value of \( x \) given that a solid metallic sphere of radius 3 cm is melted and recast into three smaller spheres with radii 1.5 cm, 2 cm, and \( x \) cm.

### Step 1: Calculate the volume of the original sphere
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

For the original sphere with radius \( r = 3 \) cm:

\[ V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the smaller spheres
Now, we will calculate the volumes of the two smaller spheres with known radii.

1. For the sphere with radius \( 1.5 \) cm:

\[ V_1 = \frac{4}{3} \pi (1.5)^3 = \frac{4}{3} \pi \left(\frac{27}{8}\right) = \frac{4 \cdot 27}{3 \cdot 8} \pi = \frac{36}{8} \pi = 4.5 \pi \text{ cm}^3 \]

2. For the sphere with radius \( 2 \) cm:

\[ V_2 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

### Step 3: Set up the equation for the total volume
The total volume of the three smaller spheres must equal the volume of the original sphere:

\[ V_1 + V_2 + V_3 = V \]

Substituting the known values:

\[ 4.5 \pi + \frac{32}{3} \pi + V_3 = 36 \pi \]

### Step 4: Solve for \( V_3 \)
First, we need to express \( 4.5 \) as a fraction:

\[ 4.5 = \frac{9}{2} \]

Now, we find a common denominator for \( \frac{9}{2} \) and \( \frac{32}{3} \). The least common multiple of 2 and 3 is 6.

Converting \( \frac{9}{2} \) and \( \frac{32}{3} \) to have a denominator of 6:

\[ \frac{9}{2} = \frac{27}{6}, \quad \frac{32}{3} = \frac{64}{6} \]

Now we can add these:

\[ \frac{27}{6} + \frac{64}{6} = \frac{91}{6} \]

Now substituting back into the equation:

\[ \frac{91}{6} + V_3 = 36 \]

### Step 5: Isolate \( V_3 \)
To isolate \( V_3 \), we subtract \( \frac{91}{6} \) from \( 36 \):

\[ V_3 = 36 - \frac{91}{6} \]

Converting \( 36 \) to have a denominator of 6:

\[ 36 = \frac{216}{6} \]

Now, we can perform the subtraction:

\[ V_3 = \frac{216}{6} - \frac{91}{6} = \frac{125}{6} \text{ cm}^3 \]

### Step 6: Find the radius \( x \) of the third sphere
Using the volume formula for the sphere again:

\[ V_3 = \frac{4}{3} \pi x^3 \]

Setting this equal to \( \frac{125}{6} \):

\[ \frac{4}{3} \pi x^3 = \frac{125}{6} \]

### Step 7: Solve for \( x^3 \)
Multiply both sides by \( \frac{3}{4\pi} \):

\[ x^3 = \frac{125}{6} \cdot \frac{3}{4\pi} = \frac{375}{24\pi} \]

### Step 8: Calculate \( x \)
Taking the cube root:

\[ x = \sqrt[3]{\frac{375}{24\pi}} \]

This value can be approximated numerically if needed.

### Final Answer
The value of \( x \) is approximately \( 1.5 \) cm.

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