Question

the curved surface of a solid metallic sphere is cut in such a way that the area of the new sphere is half of the previous one. let us calculate the ratio of tev of the portion cut off and the remaining portion of the sphere. volumes

Answer 1

Given:

The curved surface of a solid metallic sphere is cut in such a way that the area of the new sphere is half of the previous one.

To find:

The ratio of the volumes of the portion cut off and the remaining portion of the sphere.

Solution:

We know,

$\boxed{\bold{Surface\:area \:of\:a\:sphere = 4 \pi r^2}}\\\\\boxed{\bold{Volume \:of\:a\:sphere = \frac{4}{3} \pi r^2}}$

The curved surface area of the solid metallic sphere = 4πr²

The original sphere is cut in such a way that the area of the new sphere is half of the previous one, therefore, we have

The curved surface area of each of the new sphere = ½ × 4πr² = 2πr²

So, we can say that,

→ The new spheres so received after cutting the original sphere are hemispheres

→ Both the hemispheres have equal C.S.A.

→ Both the hemispheres have the same radius

∴ The volume of each of the hemispheres = $\frac{4}{3} \pi r^2$

Now,

The ratio of the volumes of the portion cut off and the remaining portion of the sphere is,

= $\frac{\frac{4}{3} \pi r^2}{\frac{4}{3} \pi r^2}$

both have the same radius  

= 1 : 1

Thus, the ratio of the volumes of the portion cut off from the sphere and the remaining portion of the sphere is → 1 : 1.

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Answer 2

Answer:

$\huge\mathfrak\pink{✩Answer☆}$

Let the radius of the old sphere be =R unit

let the radius of the new sphere be =r unit

therefore,curved surface area of the old sphere =4πR²

and the curved surface area of the new sphere =4πr²

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ATP,

4πR²/2=4πr²

or,R²=2r²

or ,R²=√2r²

or,R²=√2r

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now,volume of the old sphere=4/3πr³=4/3(2r)³ cubic unit

volume of the new sphere=4/3πr³

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volume of the remaining sphere =4/3(√2 r)³-4/3 πr³

⠀⠀ ⠀⠀ ⠀ ⠀⠀ ⠀⠀ ⠀⠀ ⠀⠀ ⠀ =4/3π³(2√2-1)

therefore,the ratio of the cut off portion and remaining part =4/3πr³:4/3πr³(2√2-1)

=1:2√2-1 (ANS)

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hope this helps you.