Question

A cylindrical container of 36 cm height and 48 cm diameter is filled with sand. Now, its sand is used to form a conical heap of radius 30 cm. The height (in cm) of the conical heap is:

• A. 67.96
• B. 68.84
• C. 69.12
• D. 70.22

Hint:

For such volume problems, remember that the volume of a cone is \( \frac{1}{3} \pi r^2 h \), and the volume of a cylinder is \( \pi r^2 h \). Set the volumes equal when the material is conserved (no sand is lost).

Answer 1

The Correct Option is C

The volume of sand in the cylindrical container is given by: \[ \text{Volume of cylinder} = \pi r^2 h = \pi \times \left( \frac{48}{2} \right)^2 \times 36 = \pi \times 24^2 \times 36 = 20736\pi \, \text{cm}^3 \] Now, the sand is used to form a conical heap. The volume of the cone is given by: \[ \text{Volume of cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 30^2 \times h = 300\pi h \, \text{cm}^3 \] Since the volume of sand is the same in both cases, we equate the volumes: \[ 20736\pi = 300\pi h \] Canceling \( \pi \) from both sides: \[ 20736 = 300h \] \[ h = \frac{20736}{300} = 69.12 \, \text{cm} \] Thus, the height of the conical heap is 69.12 cm.