A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.

Answer

To solve the problem of finding the number of smaller cones formed from a solid metallic sphere, we will follow these steps:

### Step 1: Calculate the Volume of the 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. Here, the radius \( r \) is 10.5 cm.

Substituting the value:
\[ V = \frac{4}{3} \pi (10.5)^3 \]

### Step 2: Calculate \( (10.5)^3 \)
Calculating \( (10.5)^3 \):
\[ 10.5 \times 10.5 = 110.25 \]
\[ 110.25 \times 10.5 = 1157.625 \]
Thus, \( (10.5)^3 = 1157.625 \).

### Step 3: Calculate the Volume of the Sphere
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (1157.625) \]
\[ V = \frac{4 \times 1157.625}{3} \pi = \frac{4630.5}{3} \pi \approx 1543.5 \pi \text{ cm}^3 \]

### Step 4: Calculate the Volume of One Cone
The formula for the volume \( V_c \) of a cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone. Here, the radius \( r \) is 3.5 cm and the height \( h \) is 3 cm.

Substituting the values:
\[ V_c = \frac{1}{3} \pi (3.5)^2 (3) \]

### Step 5: Calculate \( (3.5)^2 \)
Calculating \( (3.5)^2 \):
\[ 3.5 \times 3.5 = 12.25 \]
Thus, \( (3.5)^2 = 12.25 \).

### Step 6: Calculate the Volume of One Cone
Now substituting back into the volume formula for the cone:
\[ V_c = \frac{1}{3} \pi (12.25)(3) \]
\[ V_c = \frac{1}{3} \pi (36.75) = 12.25 \pi \text{ cm}^3 \]

### Step 7: Set the Volume of the Sphere Equal to the Total Volume of Cones
Let \( n \) be the number of cones formed. The total volume of \( n \) cones is:
\[ n \cdot V_c = n \cdot (12.25 \pi) \]
Setting the volume of the sphere equal to the total volume of the cones:
\[ 1543.5 \pi = n \cdot (12.25 \pi) \]

### Step 8: Solve for \( n \)
Dividing both sides by \( \pi \):
\[ 1543.5 = n \cdot 12.25 \]
Now, solving for \( n \):
\[ n = \frac{1543.5}{12.25} \approx 126 \]

### Conclusion
The number of smaller cones formed is \( n = 126 \).

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