In a right circular cone, the radius of its base is 7 cm and its height is 24 cm. A cross-section is made through the mid-point of the height parallel to the base. The volume of the upper portion is :

Answer: Option A

Solution

Answer: Option A

Solution:

$$\begin{array}{rl}& r=7\text{ cm, }h\text{ = 24 cm}\\ & \text{Now, }△\text{AOB}\sim △\text{COD}\\ & \text{So, }\frac{OA}{OC}=\frac{AB}{CD}\\ & \Rightarrow \frac{h}{\frac{h}{2}}=\frac{r}{CD}\\ & \Rightarrow CD=\frac{r}{2}\end{array}$$
∴ Volume of upper portion :
$$\begin{array}{rl}& =\frac{1}{3}\pi {\left(\frac{r}{2}\right)}^2\left(\frac{h}{2}\right)\\ & =\left(\frac{1}{3}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\times 12\right)\text{ c}{\text{m}}^3\\ & =154\text{ c}{\text{m}}^3\end{array}$$