Three spherical balls of radius 3 cm, 2 cm and 1 cm are melted to form a new spherical ball. In this prcoess there is a loss of 25% of the material. What is the radius (in cm) of the new ball ?

Options

A. 5
B. 3
C. 6
D. 8

Answer

To find the radius of the new spherical ball formed by melting three smaller balls with a loss of 25% of the material, we can follow these steps:

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

1. For the first sphere with radius \( r_1 = 3 \) cm:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

2. For the second sphere with radius \( r_2 = 2 \) cm:
\[ V_2 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

3. For the third sphere with radius \( r_3 = 1 \) cm:
\[ V_3 = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi (1) = \frac{4}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the total volume of the three spheres
Now, we add the volumes of the three spheres:
\[ V_{total} = V_1 + V_2 + V_3 = 36 \pi + \frac{32}{3} \pi + \frac{4}{3} \pi \]
To add these, we convert \( 36 \pi \) into a fraction:
\[ 36 \pi = \frac{108}{3} \pi \]
Now, we can add:
\[ V_{total} = \frac{108}{3} \pi + \frac{32}{3} \pi + \frac{4}{3} \pi = \frac{144}{3} \pi = 48 \pi \text{ cm}^3 \]

### Step 3: Account for the loss of material
Since there is a loss of 25% of the material, we only retain 75% of the total volume:
\[ V_{new} = 75\% \text{ of } V_{total} = 0.75 \times 48 \pi = 36 \pi \text{ cm}^3 \]

### Step 4: Set up the equation for the new sphere
Let \( r \) be the radius of the new sphere. The volume of the new sphere is:
\[ V_{new} = \frac{4}{3} \pi r^3 \]
Setting this equal to the volume we calculated:
\[ \frac{4}{3} \pi r^3 = 36 \pi \]

### Step 5: Solve for \( r^3 \)
Dividing both sides by \( \pi \):
\[ \frac{4}{3} r^3 = 36 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ r^3 = 36 \times \frac{3}{4} = 27 \]

### Step 6: Find the radius \( r \)
Taking the cube root of both sides:
\[ r = \sqrt[3]{27} = 3 \text{ cm} \]

Thus, the radius of the new ball is **3 cm**.

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