Question

A sphere and a cone have the same radius. If the volume of the sphere is four times
of the volume of the cone, find the ratio of the height and radius of the cone.​

Answer 1

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Step-by-step explanation:

The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h.

Answer 2

Solution :

It is given that ;

$\sf{Radius _{(Cone)} = Radius _{(Sphere)} }$

And also volume of the sphere is equals to 4 times of volume of the cone which means ;

$\sf{Volume_{(sphere)} = 4×Volume_{(cone)}} \ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: : - (1)$

Now Using Formulas to calculate volumes :

$\boxed{\sf{Volume_{(cone)}} = \frac{1}{3}\pi {r}^{2}h} \\ \\ \\ \boxed{ \sf{Volume_{(sphere) = \: \dfrac{4}{3}\pi {r}^{3} } }}$

$\: \: \: \: \: \: \: \sf{Height_{(\:cone)} : Radius_{(sphere)}}$

We can take radius of sphere instead of cone because it is given that both radius are equal

By substituting the values we get ;

$\: \: \: \: \: \sf{ = \dfrac{ \dfrac{1}{3}\pi {r}^{2}h × 4 }{ \dfrac{4}{3} \pi {r}^{3} } } \\ \\ \: \: \: \: \: \sf{ = \cancel{ \dfrac{ \dfrac{4}{3}\pi \times r \times r \: \times h }{ \dfrac{4}{3} \pi \times {r} \times r \times r } }} \\ \\ \sf{ = {\dfrac{h}{r}} } \: \\ \\ \sf{ = h= r}\\\\ \sf{1: \: 1}$

Therefore , ratio of the volume of come and sphere is equals to 1 : 1