How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm?

Answer

To find out how many spherical lead shots of diameter 4 cm can be made from a solid cube of lead with an edge measuring 44 cm, we will follow these steps:

### Step 1: Calculate the volume of the cube.
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of the edge of the cube.

For our cube:
\[ a = 44 \, \text{cm} \]
So, the volume of the cube is:
\[ V = 44^3 = 44 \times 44 \times 44 \]

### Step 2: Calculate the volume of one spherical lead shot.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given the diameter of the lead shot is 4 cm, the radius \( r \) is:
\[ r = \frac{4}{2} = 2 \, \text{cm} \]
Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi \times 8 = \frac{32}{3} \pi \, \text{cm}^3 \]

### Step 3: Set up the equation to find the number of spherical lead shots.
Let \( N \) be the number of spherical lead shots that can be made. The total volume of the spherical shots will equal the volume of the cube:
\[ \text{Volume of cube} = N \times \text{Volume of one sphere} \]
This can be expressed as:
\[ 44^3 = N \times \frac{32}{3} \pi \]

### Step 4: Solve for \( N \).
Rearranging the equation gives:
\[ N = \frac{44^3}{\frac{32}{3} \pi} \]
This simplifies to:
\[ N = \frac{44^3 \times 3}{32 \pi} \]

### Step 5: Calculate \( N \).
First, calculate \( 44^3 \):
\[ 44^3 = 44 \times 44 \times 44 = 85184 \, \text{cm}^3 \]
Now substituting this value back into the equation for \( N \):
\[ N = \frac{85184 \times 3}{32 \pi} \]
Calculating \( 85184 \times 3 = 255552 \).

Now, we need to divide by \( 32 \pi \):
\[ N = \frac{255552}{32 \pi} \]
Using \( \pi \approx 3.14 \):
\[ N \approx \frac{255552}{32 \times 3.14} \approx \frac{255552}{100.48} \approx 2541.67 \]
Since \( N \) must be a whole number, we take the floor value:
\[ N \approx 2541 \]

### Conclusion:
Thus, the number of spherical lead shots that can be made from the solid cube of lead is approximately **2541**.

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