Question

From a solid metallic cylinder with height 10 cm and radius of base 6 cm, a right circular cone of the same height and same base is removed. Find the volume and the surface area of the remaining solid.

Answer 1

Given:

Solid cylinder: radius r = 6 cm, height h = 10 cm

A right circular cone of same radius and same height is removed.
Use `π = 22/7`

(i) Volume of remaining solid

Step 1: Volume of cylinder

`V_"cyl" ​= πr^2h`

`22/7 xx 6^2 xx 10`

= `22/7 xx 36 xx 10`

= `7920/7`

= 1131.43 cm³

Step 2: Volume of cone removed

`V_"cone" ​= 1/3 πr^2h`

= `1/3 xx 22/7 xx 36 xx 10`

= `1/3 xx 1131.43`

= 377.14 cm³

Step 3: Volume of remaining solid

V = `V_"cyl"​ − V_"cone"`​

= 1131.43 − 377.14

= 754.29 cm³

V = 754.29 cm^3​

(ii) Surface area of the remaining solid

`CSA_"cyl"​ = 2πrh`

= `2 × 22/7 xx 6 xx 10`

= `2640/7`

= 377.14 cm²

Step 2: Area of the bottom base

`πr^2 = 22/7 xx 6^2`

`22/7 xx 36 = 792/7`

= 113.14 cm²

Step 3: Slant height of the cone

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

`l = sqrt(6^2 + 10^2)`

`l = sqrt(136)`

`l = 11.66`

Step 4: Curved surface area of a conical hollow

`CSA_"cone" ​= πrl`

= `22/7 xx 6 xx 11.66`

= 219.88 cm²

Step 5: Total surface area of the remaining solid

TSA = 377.14 + 113.14 + 219.88

= 710.16 cm²

Surface Area = 710.16 cm²