Question

একটি লম্ব বৃত্তাকার শঙ্কুর পার্শ্বতলের ক্ষেত্রফল
তার ভূমির ক্ষেত্রফলের √17 গুণ হলে শঙ্কুর উচ্চতা ও ভূমির
ব্যাসের অনুপাত কত হবে?​

Answer 1

❏ Formula Used:-

For a right circular cone of base radius r and height h ,

then..

$\sf\longrightarrow\boxed{ L.S.A.=\pi \times r\times l }$

AND

$\sf\longrightarrow \boxed{ \sf B.A.= \pi r{}^{2}}$

$\sf\longrightarrow \boxed{\sf Volume=\frac{1}{3}\pi r{}^{2}h}$

AND

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

where, L.S.A=curved surface area.

B.A=base area .

❏ Solution:-

Q)

For a right circular cone the curved surface area is of √17 than the Base area of the cone . Now find the ratio between the radius to the height of the cone.

Ans)

Let, the radius and the height of the cone is

r and h respectively.

∴slant height (l)=$\sqrt{r{}^{2}+h{}^{2}}$

Now ...the curved surface area of the cone ..

∴L.S.A.=πrl

And the Base area

∴B.A=πr²

Now , according to the Question.

➾the curved surface area is of √17 than the Base area.

i.e.,

➾ L.S.A=√17(B.A.)

➾(L.S.A)²=[√17(B.A)]²

➾(L.S.A)=17(B.A.)²

(putting the values)

➾(πrl)²=17×(πr²)

➾π²r²l²=17×π²r²×r²

➾l²=17r² (cancelling π² and r² from both sides)

➾r²+h²=17r²

➾17r²-r²=h²

➾16r²=h²

➾(4r)²=h²

➾4r=h

$\sf\longrightarrow \Large{\frac{r}{h}=\frac{1}{4}}$

$\sf\longrightarrow \huge{\boxed{\red{r : h=1 : 4}}}$

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$\underline{ \huge\mathfrak{hope \: this \: helps \: you}}$

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