Question

A toy is in the form of a cone mounted on a hemisphere with the same radius. The radius of the base of the cone is 8 cm and height is 6 cm. Find the surface area and the volume of the toy. (Use π = 3.14)

Answer 1

Given:

A cone mounted on a hemisphere of the same radius.

Radius, r = 8 cm

Height of cone, h = 6 cm

π = 3.14

1) Surface Area of the Toy

(Only the outer surface is counted. The common circular base is not exposed.)

Step 1: Slant height of cone

`l = sqrt(r^2 + h^2) = sqrt(8^2 + 6^2)`

`l = sqrt(64 + 36)`

`l = sqrt100`

l = 10 cm

Step 2: Curved surface area (CSA) of a cone

CSA_cone​ = πrl = 3.14 × 8 × 10 = 251.2 cm²

Step 3: Curved surface area (CSA) of a hemisphere

`CSA_"hemi" ​= 2πr^2`

= 2 × 3.14 × 8²

= 2 × 3.14 × 64 = 401.92 cm²

Step 4: Total surface area 

Surface Area = 251.2 + 401.92 = 653.12 cm

Surface Area of toy = 653.12 cm²

2) Volume of the Toy

Step 1: Volume of cone

`V_"cone" = 1/3 πr^2h`

= `1/3 xx 3.14 xx 8^2 xx 6`

= `1/3 xx 3.14 xx 64 xx 6`

= `1/3 xx 1205.76`

= 401.92 cm³

Step 2: Volume of hemisphere

`V_"hemi​" =2/3 πr^3`

= `2/3 xx 3.14 xx 8^3`

`2/3 xx 3.14 xx 512`

= `2/3 xx 1607.68`

= 1071.79 cm³

Step 3: Total volume

Volume = 401.92 + 1071.79

= 1473.71 cm³

Volume of toy =1473.71 cm³