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

Identify the Given Dimensions

Hollow Hemisphere:

Internal radius (r) = `5sqrt2` cm

External radius (R) = 10 cm

Height (h) = `5sqrt7` cm

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

`l = sqrt((5sqrt2)^2 + (5sqrt7)^2)`

`l = sqrt((25xx2) + (25xx7))`

`l = sqrt(50 + 175)`

`=sqrt225`

= 15 cm

Curved Surface Area (CSA) of the outer hemisphere: 2πR²

Area of the rim = πR² - πr²

Total surface area = Outer CSA of hemisphere + CSA of cone + Area of rim

`TSA = 2piR^2 + pir_cl + (piR^2 - pir^2)`

`TSA = 3piR^2 + pirl - pir^2`

Now, let’s plug in our numbers:

`TSA = pi [3(10)^2 + (5sqrt2)(15) - (5sqrt2)^2]`

`TSA = pi [3(100) + 75sqrt2 - 50]`

`TSA = pi [300+75 sqrt2]`

`TSA = pi [250 + 75sqrt2]`

`TSA = pi [250 + 75(1.4)]`

`TSA = pi[250 + 105]`

`TSA = 355 pi cm^2`