Question

A juice seller was serving his customers using glasses as shown in the given figure. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of the glass was 10 cm, then find the apparent capacity of the glass and its actual capacity. (Take } $\pi = 3.14$ \textbf{) 

Hint:

Always subtract the volume of the raised (or hollow) part from the main body to find the actual capacity.

Answer 1

Step 1: Given data.
Diameter of the glass = 5 cm \[ \Rightarrow \text{Radius (r)} = \dfrac{5}{2} = 2.5 \, \text{cm} \] Height of the glass = 10 cm

Step 2: Apparent capacity (volume of full cylinder).
\[ V_{\text{cylinder}} = \pi r^2 h \] \[ V_{\text{cylinder}} = 3.14 \times (2.5)^2 \times 10 = 3.14 \times 6.25 \times 10 = 196.25 \, \text{cm}^3 \]
Step 3: Volume of the hemispherical raised portion (to be subtracted).
\[ V_{\text{hemisphere}} = \dfrac{2}{3} \pi r^3 \] \[ V_{\text{hemisphere}} = \dfrac{2}{3} \times 3.14 \times (2.5)^3 = \dfrac{2}{3} \times 3.14 \times 15.625 = 32.67 \, \text{cm}^3 \]
Step 4: Actual capacity of the glass.
\[ V_{\text{actual}} = V_{\text{cylinder}} - V_{\text{hemisphere}} = 196.25 - 32.67 = 163.58 \, \text{cm}^3 \] Step 5: Conclusion.
Apparent capacity = 196.25 cm$^3$ Actual capacity = 163.58 cm$^3$