How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm?
2541 shots
Step 1: Find the volume of the cube.
The formula for the volume of a cube is:
\( V = a^3 \), where \(a\) is the length of the edge.
Here, \( a = 44\,\text{cm} \).
So, volume of cube = \( 44^3 = 85184\,\text{cm}^3 \).
Step 2: Find the volume of one spherical lead shot.
We are told the diameter of each spherical shot = 4 cm.
So, radius \( r = \dfrac{4}{2} = 2\,\text{cm} \).
Formula for volume of a sphere is:
\( V = \dfrac{4}{3} \pi r^3 \).
Substitute \( r = 2 \):
\( V = \dfrac{4}{3} \pi (2)^3 = \dfrac{4}{3} \pi \times 8 = \dfrac{32}{3} \pi \).
So, the volume of one spherical shot = \( \dfrac{32}{3}\pi\,\text{cm}^3 \).
Step 3: Find the number of shots.
Total number of shots = (Volume of cube) ÷ (Volume of one sphere)
= \( \dfrac{85184}{(32/3)\pi} \).
= \( \dfrac{85184 \times 3}{32 \pi} \).
= \( \dfrac{255552}{32 \pi} \).
= \( \dfrac{7986}{\pi} \).
Now, using \( \pi \approx 3.1416 \):
\( \dfrac{7986}{3.1416} \approx 2541 \).
Final Answer:
The cube of lead can be melted to make 2541 spherical shots.