Question

a conical tent of capacity 1232m^3 stands on a circular base of area 154m^2.Find in m^2 the area of the canvas​

Answer 1

Answer: 550 m²

Given:

$\sf Capacity\:of\:conical\:tent=1232\:m^3$

$\sf Area\:of\:the\:circular\:base=154\:m^2$

To find:

$\sf Area\:of\:the\:canvas.$

Formula used:

$\boxed{\sf Volume\:of\:cone=\dfrac{1}{3}\pi r^2h}$

$\boxed{\sf Area\:of\:the\:circle=\pi r^2}$

$\boxed{\sf Area\:of\:cone=\pi rl}$

$\boxed{\sf Slant\:height\:of\:the\:cone(l)=\sqrt{h^2+r^2} }$

Explanation:

$\sf According\:to\:the\:question,$

$\Rightarrow\sf \dfrac{1}{3}\pi r^2h=1232\:m^3$

$\sf Substituting\:the\:value\:of\:\pi r^2h\:in\:the\:above\:equation\:we\:get:$

$\Rightarrow\sf \dfrac{1}{3}\times 154\times h=1232\:m^3$

$\Rightarrow\sf h=\dfrac{1232\times3}{154}$

$\Rightarrow\boxed{\sf h=24\:m}$

$\sf It\:is\:also\:given\:that,$

$\sf \pi r^2=154 \:m^2$

$\sf Therefore,$

$\Rightarrow\sf \dfrac{22}{7}\times r^2=154$

$\Rightarrow\sf r^2=154\times \dfrac{7}{22}$

$\Rightarrow\sf r^2=49$

$\Rightarrow\boxed{\sf r=7\:m}$

$\sf We\:know\:that,$

$\sf Area\:of\:canvas=\pi rl$

                      $\sf =\dfrac{22}{7}\times 7\times \sqrt{h^2+r^2}$

                      $\sf =22 \times \sqrt{24^2+7^2}$

                      $\sf =22\times\sqrt{625}$

                      $\sf =22\times25$

                      $\sf =\boxed{\sf 550\:m^2}$

Therefore, the area of the canvas is 550 m².

Answer 2

Answer:

$\huge{\mathcal{\underline{\green{SOLUTION }}}}$

Let r = Radius of the base of the conical tent

h = Height of the conical tent

l = Slant Height of the conical tent

So

$Area \: of \: the \: base \: = \pi {r}^{2} \: \: \: {cm}^{2}$

$Volume \: of \: the \: cone \: = \frac{1}{3} \pi{r}^{2} h \: \: {cm}^{3}$

Therefore

$\pi{r}^{2} \: \: = \: 154$

${r}^{2} = 154 \times \frac{7}{22}$

${r}^{2} = \: 49$

$r \: = \sqrt{49} = 7$

Again

$\frac{1}{3} \pi{r}^{2} h \: = 1232$

$Since\: \: \pi{r}^{2} \: \: = \: 154$

So

$\frac{1}{3} \times 154 \times h \: = 1232$

$\implies \: h \: = 1232 \times \frac{3}{154}$

$\implies \: h \: = 24$

Again

${l}^{2} = {r}^{2} + {h}^{2}$

$\implies \: {l}^{2} = {7}^{2} + {24}^{2}$

$\implies \: {l}^{2} = 625$

$\implies \: {l} = \sqrt{625} = 25$

$Hence\: the \: area \: canvas$

$= \pi \: r \: l \: \: \: \: {cm}^{2}$

$= \frac{22}{7} \times 7 \times 25 \: \: {cm}^{2}$

$= 550 \: \: {cm}^{2}$