Question

The curved surface area of a right circular cone of height 15cm and base diameter 16 cm is………………………….
$$\begin{gathered} (a)\text{ 60}\pi \text{c}{\text{m}}^2 \\ (b)\text{ 68}\pi \text{c}{\text{m}}^2 \\ (c)\text{ 120}\pi \text{c}{\text{m}}^2 \\ (d)\text{ 136}\pi \text{c}{\text{m}}^2 \end{gathered}$$

Answer 1

Hint: In this question the height and base diameter of a right circular cone is given. Use the direct formula for curved surface area for the right circular cone ($\pi rl$) to evaluate the required quantity.

Complete step-by-step answer:

As we know that the curved surface area (C.S.A) of the right circular cone is $\pi rl$.
Where, l = slant height of the cone.
              r = base radius of the cone.
Now it is given that the height (h) of the cone is 15 cm and the diameter (d) of the cone is 16 cm.
Now as we know that the radius (r) is half of the diameter (d).
$\Rightarrow r={\displaystyle \frac{d}{2}}={\displaystyle \frac{16}{2}}=8$ cm.
Now in a right circular cone slant height (l) is given as
In triangle ABC apply Pythagoras theorem we have,
$\Rightarrow l^2=h^2+r^2$
$\Rightarrow l=\sqrt{h^2+r^2}$
So substitute the value in the above equation we have,
$\Rightarrow l=\sqrt{{\left(15\right)}^2+8^2}=\sqrt{225+64}=\sqrt{289}=17$ cm.
So the curved surface area (C.S.A) of the right circular cone is
 $\Rightarrow C.S.A=\pi rl=\pi \left(8\right)\left(17\right)=136\pi \text{ c}{\text{m}}^2$.
So this is the required curved surface area (C.S.A) of a right circular cone.
Hence option (d) is correct.

Note: Whenever we face such types of problems the key point is to have the good gist of the diagrammatic representation using the information provided in the question as it helps understanding the geometry of the figure involved. It is always advisable to learn the direct formula for some of the basic conic sections like the right circular cone as it helps to save a lot of time.