Question

A solid is in the shape of a cone mounted on a hemisphere of same base and radius. If the curved surface areas of the hemispherical part and the conical part are equal then find the ratio of the radius and the height of the conical part.​

Answer 1

✬ Required Ratio = 1 : √3 ✬

Step-by-step explanation:

Given:

• Radius of hemisphere and cone are same.
• Curved surface areas of conical and hemispherical part are equal.

To Find:

• Ratio of the radius and the height of the conical part.

Solution: Since both areas are equal to each other.

As we know that

★ CSA of Hemisphere = 2πr² ★

★ CSA of cone = πrl ★

A/q

• 2πr² = πrl

Also we know that slant height (l) = √h² + r²

➮ 2πr² = πrl

➮ 2πr² = πr√h² + r²

➮ 2r² = r√h² + r²

➮ 2r = √h² + r²

Squaring both sides

$\implies{\rm }$ (2r)² = (√h² + r²)²

$\implies{\rm }$ 4r² = h² + r²

$\implies{\rm }$ 4r² – r² = h²

$\implies{\rm }$ 3r² = h²

$\implies{\rm }$ r²/h² = 1/3

$\implies{\rm }$ r/h = √1/3

$\implies{\rm }$ r/h = 1/√3

$\implies{\rm }$ r : h = 1 : √3

Hence, ratio of radius of hemispherical part and the height of conical part is 1 : √3.

Answer 2

Answer:

✳️ Given ✳️

• A solid is in the shape of a cone mounted on a hemisphere of same base and radius.
• Curved surface area of the hemisphere part and the conical part are equal.

✳️ To Find ✳️

• What is the ratio of the radius and the height of the conical part.

✳️ Formula Used ✳️

$\green\bigstar$ CSA Of Hemisphere = 2πr²

$\red\bigstar$ CSA Of Cone = πrl

✳️ Solution ✳️

✒️ Radius of hemisphere and cone = r

✒️ Slant height of the cone = l

✒️ Curved surface area of hemisphere = 2πr²

✒️ Curved surface of the cone = πrl

➕ According to the question,

$\Rightarrow$ πrl = 2πr²

$\implies$ l = 2r

$\implies$ $\sqrt{h² + r²}$ = 2r

✍️ Squaring both sides we get,

$\Rightarrow$ h² + r² = 4r²

$\implies$ 3r² = h²

$\implies$ $\dfrac{r²}{h²}$ = $\dfrac{1}{3}$

$\implies$ $\dfrac{r}{h}$ = $\dfrac{1}{\sqrt{3}}$

$\dashrightarrow$ r : h = 1 : $\sqrt{3}$

$\therefore$ Ratio of the radius and the height of the conical part are 1 : $\sqrt{3}$

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