How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece of dimensions 66 cm, 42 cm and 21 cm?
1500 shots
Step 1: Find the volume of the rectangular lead block.
Length = 66 cm, Breadth = 42 cm, Height = 21 cm.
Volume of block = \(66 \times 42 \times 21 = 58,212\,\text{cm}^3\).
Step 2: Find the radius of one spherical shot.
Diameter of one shot = 4.2 cm.
Radius = Diameter ÷ 2 = \(4.2 \div 2 = 2.1\,\text{cm}\).
Step 3: Find the volume of one spherical shot.
Formula: Volume of sphere = \(\dfrac{4}{3} \pi r^3\).
Here, \(r = 2.1\,\text{cm}\).
So, Volume = \(\dfrac{4}{3} \pi (2.1)^3\).
First calculate \((2.1)^3 = 2.1 \times 2.1 \times 2.1 = 9.261\).
Now, Volume = \(\dfrac{4}{3} \pi \times 9.261 = 38.808 \pi / 3 = 12.936 \pi\,\text{cm}^3\).
In fraction form: \(\dfrac{6174}{500} \pi\,\text{cm}^3\).
Step 4: Find how many such spheres can be made from the block.
Number of shots = (Volume of block) ÷ (Volume of one shot).
\(= \dfrac{58,212}{(6174/500)\pi}\).
Simplify: \(= \dfrac{58,212 \times 500}{6174 \pi}\).
\(= \dfrac{29,106,000}{6174 \pi}\).
\(= \dfrac{4715}{\pi}\).
Using \(\pi = 3.1416\), we get approximately 1500.
Final Answer: 1500 spherical lead shots can be obtained.