How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm \(\times\) 11 cm \(\times\) 12 cm?
84 shots
Step 1: Find the volume of the cuboid (the big lead block).
Formula: \(\text{Volume of cuboid} = \text{length} \times \text{breadth} \times \text{height}\)
Here: \(9\,\text{cm} \times 11\,\text{cm} \times 12\,\text{cm} = 1188\,\text{cm}^3\).
Step 2: Find the radius of one spherical shot.
Diameter = 3 cm, so Radius = \(\dfrac{3}{2} = 1.5\,\text{cm}\).
Step 3: Find the volume of one spherical shot.
Formula: \(\text{Volume of sphere} = \dfrac{4}{3} \pi r^3\).
Substitute: \(\dfrac{4}{3} \pi (1.5)^3 = \dfrac{4}{3} \pi (3.375) = 4.5 \pi\,\text{cm}^3\).
Step 4: Find the number of spherical shots.
Formula: \(\text{Number of shots} = \dfrac{\text{Volume of cuboid}}{\text{Volume of one sphere}}\).
Substitute: \(\dfrac{1188}{4.5\pi} = \dfrac{1188}{\dfrac{99}{7}} = \dfrac{1188 \times 7}{99} = 84\).
Final Answer: 84 shots can be made.