A cylinder and a cone are of the same base radius and of same he
The radius of a solid hemispherical toy is 3.5 cm. Find the total surface area.

Here, we have
radius of hemispherical toy (r) = 3.5 cm
Now,
Total surface area = 3πr2

equals space open parentheses 3 space straight x space 22 over 7 straight x space 3.5 space straight x space 3.5 close parentheses space space cm squared
equals space 115.5 space cm squared

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If the number of square centimetres on the surface of a sphere is equal to the number of cubic centimetres in its volume, what is the diameter of the sphere?

It is given that
Volume of sphere = Surface Area of sphere

rightwards double arrow space 4 over 3 πr cubed equals 4 πr squared
rightwards double arrow space 3 space equals space straight r space or space straight r space equals space 3 space cm
rightwards double arrow space space and space diameter space equals space 2 space straight x space 3 space equals space 6 space cm

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A right circular cylinder and a right circular cone have the same base radius and same height. What is the ratio of the volume of the cylinder to that of cone?

Let V and ‘R’ be the radii of cylinder and cone respectively.
And ‘h’ and ‘H’ be the heights of cylinder and cone respectively.

Now volume of cylinder (v1) = πr squared straight h
And, volume of cone (v2) = 1 third space πr squared straight h
Now comma space space space space straight v subscript 1 over straight v subscript 2 equals fraction numerator πr squared straight h over denominator begin display style 1 third end style πR squared straight H end fraction
space space space space space space space space space space space space space space space space equals space fraction numerator πr squared straight h over denominator begin display style 1 third end style πr squared straight h end fraction space left square bracket because space straight r space equals space straight R comma space straight h space equals space straight H right square bracket
space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator begin display style 1 third end style end fraction equals 3 over 1

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Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.

Let V1 and V2 be the volume of the cube and sphere respectively, then
V1= (2r)3 and V24 over 3 comma space πr cubed
therefore space space space space straight V subscript 1 over straight V subscript 2 equals fraction numerator left parenthesis 2 straight r right parenthesis cubed over denominator begin display style 4 over 3 end style πr cubed end fraction equals fraction numerator 8 straight r cubed over denominator begin display style 4 over 3 end style πr cubed end fraction equals 6 over straight pi
Hence comma space straight V subscript 1 space colon space straight V subscript 2 equals 6 space colon space straight pi
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A cylinder and a cone are of the same base radius and of same height. Find the ratio of the volume of cylinder to that of the cone.


Let r be the base radius and h be the height.
Then, volume of the cylinder i. e.,
V1 = πr2h
and volume of the cone i.e.,

straight V subscript 2 equals 1 third πr squared straight h
therefore space space straight V subscript 1 over straight V subscript 2 equals fraction numerator πr squared straight h over denominator 1 divided by 3 space πt squared straight h end fraction equals 3 over 1

Hence, the required ratio is 3 : 1.

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