Question

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1. A conical tent requires 264 m² of canvas. If the slant
height is 12 m, find the vertical height.
​

Answer 1

$\underline{\bigstar{\sf \ Given :- }}$

• C.S.A of tent = 264m^2
• Slant height (l)= 12m

$\underline{\bigstar{\sf \ To \ Find :- }}$

• We have to find the Vertical height of the tent (h)

Solution :-

• Find the value of r

$\boxed{\sf \dag\ C.S.A \ of \ cone = \pi r l }$

$\dashrightarrow\sf \pi r l = 264\\ \\ \\ \dashrightarrow\sf \dfrac{22}{7}\times 12\times r = 264\\ \\ \\ \dashrightarrow\sf 12r= \dfrac{264\times 7}{\cancel{22}}\\ \\ \\ \dashrightarrow\sf r= \dfrac{\cancel{12}\times 7}{\cancel{12}}\\ \\ \\ \dashrightarrow{\sf {\red{\ r= 7\ m}}}$

Now we know the formula to find the slant height of the cone -

$\boxed{\sf l^2=r^2+h^2}$

$\dashrightarrow\sf (12)^2= (7)^2+h^2 \\ \\ \\ \dashrightarrow\sf 144=49+h^2\\ \\ \\ \dashrightarrow\sf h^2= 144-49\\ \\ \\ \dashrightarrow\sf h^2= 95\\ \\ \\ \dashrightarrow\sf h= \sqrt{95}\\ \\ \\ \dashrightarrow{\boxed{\sf {\star\ \ h= 9.75m}}}$

$\boxed{\bigstar{\textsf{ Vertical\ Height \ of \ cone = {\textbf{ 9.75m}}}}}$

$\underline{\bigstar{\sf \ Other\ important \ Formulas\ :- }}$

$\boxed{\bigstar{\sf \ Cylinder :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Cylinder= \pi r^2 h \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ cylinder= 2\pi r h\\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ cylinder= 2\pi r (h+r)$

$\boxed{\bigstar{\sf \ Cone :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Cone= \dfrac{1}{3}\pi r^2 h \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ Cone = \pi r l \\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ Cone = \pi r (l+r) \\ \\ \\ \sf {\textcircled{\footnotesize4}} Slant \ Height \ of \ cone (l)= \sqrt{r^2+h^2}$

$\boxed{\bigstar{\sf \ Hemisphere :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Hemisphere= \dfrac{2}{3}\pi r^3 \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ Hemisphere = 2 \pi r^2 \\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ Hemisphere = 3 \pi r^2$

$\boxed{\bigstar{\sf \ Sphere :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Sphere= \dfrac{4}{3}\pi r^3 \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Surface\ Area \ of \ Sphere = 4 \pi r^2$

Answer 2

$\small\sf\underline\blue{Question:-}$

$\rightarrow$A conical tent requires 264 m² of canvas. If the slant

height is 12 m, find the vertical height.

$\small\sf\underline\blue{Given:-}$

$\rightarrow$A conical tent requires 264 m² of canvas.

$\rightarrow$CSA of tent = 264m²

$\rightarrow$Slant height = 12m

$\rightarrow$πrl = 264

$\rightarrow$22\7×r×12 = 264

$\rightarrow$r =$264×7\div22×12$

r = 7cm

$\small\sf\underline\blue{Let \;h \:be\: the\: vertical\: height\:}$

$\small\sf\underline\blue{We \:know,}$

$\rightarrow$ l² = r²+h²

$\rightarrow$ h =√l² -√r²

$\rightarrow$ h = √12²-√7²

$\rightarrow$ h = √95

$\rightarrow$ h = 9.75 m

${\huge{\boxed{\overline{\mid{\blue{9.75}}}}}}$