Question

The internal and external radii of a hollow hemisphere are `5sqrt(2)` cm and 10 cm respectively. A cone of height `5sqrt(7)` cm and radius `5sqrt(2)` cm is surmounted on the hemisphere as shown in the figure. Find the total surface area of the object in terms of π. (Use `sqrt(2) = 1.4`)

Answer 1

Given: External radius R = 10 cm, Internal radius `r = 5sqrt(2)` cm, Cone height `h = 5sqrt(7)` cm, Cone radius = `5sqrt(2)` cm.

1. Slant height of cone:

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

`l = sqrt((5sqrt(7))^2 + (5sqrt(2))^2`

= `sqrt(175 + 50)`

= `sqrt(225)`

= 15 cm

2. Exposed Surfaces:

CSA of cone: πrl 

= `π xx 5sqrt(2) xx 15`

= `75sqrt(2)π cm^2`

Using `sqrt(2) = 1.4`:

75 × 1.4π = 105π cm²

Area of flat ring: π(R² – r²)

`π(10^2 - (5sqrt(2))^2)` 

= π(100 – 50)

= 50π cm²

Outer CSA of hemisphere: 2πR²

= 2π(100)

= 200π cm²

Inner CSA of hemisphere: 2πr²

= 2π(50) 

= 100π cm²

Total Surface Area = 105π + 50π + 200π + 100π

= 455π cm²

The total surface area of the object is 455π cm².