Question

the slant height of a cone is increased by 10%. if the radius remians the same. then the curved surface area increased by ?

Answer 1

Let r = radius of the base of the cone
Let l = slant height of the cone

$$\begin{gathered} \mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cone}=\mathrm{\pi }r\mathit{}l \\ \mathrm{When}\mathrm{the}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{cone}\mathrm{is}\mathrm{increased}\mathrm{by}10\%,\mathrm{then} \\ \mathrm{new}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{cone}=\mathit{}{l}_{\mathit{1}}\mathit{}=\mathit{}l\mathit{}+10\%\mathrm{of}l\mathit{}=\mathit{}l+\frac{l}{10}=\frac{11l}{10} \\ \mathrm{Now},\mathrm{new}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cone},r_1=\mathit{}r\left(\mathrm{given}\right) \\ \mathrm{New}\mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cone}=\mathrm{\pi }{r}_1{l}_1\mathit{}\mathit{=}\mathit{}\mathrm{\pi }\mathit{}r\frac{11l}{10}=\frac{11}{10}\mathrm{\pi }\mathit{}r\mathit{}l \end{gathered}$$

$$\begin{gathered} \mathrm{increase}\mathrm{in}\mathrm{CSA}\mathrm{of}\mathrm{cone}=\mathrm{new}\mathrm{CSA}-\mathrm{original}\mathrm{CSA} \\ =\frac{11}{10}\mathrm{\pi }r\mathit{}l-\mathrm{\pi }r\mathit{}l\mathit{}=\frac{\mathrm{\pi }r\mathit{}l}{10} \\ \mathrm{increase}\%=\left[\frac{\mathrm{increase}\mathrm{in}\mathrm{CSA}}{\mathrm{original}\mathrm{CSA}}\times 100\right]\%=\left[\frac{\mathrm{\pi }r\mathit{}l\mathit{}/10}{\mathrm{\pi }r\mathit{}l\mathit{}}\times 100\right]\%=10\% \end{gathered}$$