The curved surface of a right circular cone of height $84$ cm and base diameter $70$ cm is

### Concept & Curved Surface Area of a Cone The curved surface area (CSA) of a cone requires the slant height ($l$), which is found using the height ($h$) and radius ($r$). $$ l = \sqrt{r^2 + h^2} $$ $$ \text{CSA} = \pi \times r \times l $$ ### Step-by-Step Solution - **Given:** - Height $h = 84$ cm. - Base diameter = $70$ cm $\implies$ Radius $r = 35$ cm. - **Calculation:** - First, find the slant height $l$: - Both $35$ and $84$ are multiples of $7$. ($35 = 7 \times 5$, $84 = 7 \times 12$). - This is a multiple of the common $5, 12, 13$ right triangle. - Therefore, $l = 7 \times 13 = 91$ cm. - Now, find the CSA: - $\text{CSA} = \frac{22}{7} \times 35 \times 91$ - Cancel $7$ with $35$: $\text{CSA} = 22 \times 5 \times 91$ - $\text{CSA} = 110 \times 91 = 10010$ cm$^2$. ### Exam Strategy & Shortcut Always look for Pythagorean triplets! Recognizing that $(35, 84)$ scales down to $(5, 12)$ saves you from calculating $\sqrt{35^2 + 84^2} = \sqrt{1225 + 7056} = \sqrt{8281}$, which is time-consuming. ### Common Pitfall Using the height ($h = 84$) instead of the slant height ($l = 91$) in the curved surface area formula. ### Final Answer Therefore, the correct answer is **10010 cm²**.