Question

The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm and the other dimensions are as shown.

Calculate :

1. the total surface area.
2. the total volume of the solid and
3. the density of the material if its total weight is 1.7 kg.

Answer 1

Diameter = 10 cm

Therefore, radius (r) = 5 cm

Height of the cone (h) = 12 cm

Height of the cylinder = 12 cm

∴ `l = sqrt(h^2 + r^2)`

= `sqrt(12^2 + 5^2)`

= `sqrt(144 + 25)`

= `sqrt(169)`

= 13 cm

i. Total surface area of the solid

= `pirl + 2pirh + 2pir^2`

= `pir(l + 2h + 2r)`

= `22/7 xx 5[13 + (2xx12) + (2 xx 5)]`

= `110/7 [13 + 24 + 10]`

= `110/7 xx 47`

= `5170/7`

= 738.57 cm² 

ii. Total volume of the solid

= `1/3pir^2h + pir^2h + 2/3pir^3`

= `pir^2 [1/3h + h + 2/3r]`

= `22/7 xx 5 xx 5[1/3 xx 12 + 12 + 2/3 xx 5]`

= `550/7 [4 + 12 + 10/3]`

= `550/7 [16 + 10/3]`

= `550/7 xx 58/3`

= `31900/21`

= 1519.0476 cm³ 

iii. Total weight of the solid = 1.7 kg

∴ Density = `"Mass"/"Volume"`

= `(1.7 xx 1000)/(1519.0476)` gm/cm³

= `(17 xx 1000 xx 10000)/(10 xx 15190476)` gm/cm³

= 1.119 gm/cm³ 

`=>` Density = 1.12 gm/cm³