Question

Fermentation tanks are designed in the form of a cylinder mounted on a cone, as shown below:

The total height of the tank is 3.3 m, and the height of the conical part is 1.2 m. The diameter of the cylindrical as well as conical part is 1 m. Find the capacity of the tank. If the level of liquid in the tank is 0.7 m from the top, find the surface area of the tank in contact with liquid.

Answer 1

d = 1 m

r = `d/2`

= `1/2` m

Height of cylinder (H) = 3.3 − 1.2

= 2.1 m

Height of cone (h) = 1.2 m

Capacity of the tank = Volume of cylinder + Volume of cone

= `πr^2H + 1/3πr^2h`

= `πr^2 (H + 1/3 h)`

= `22/7 xx 1/2 xx 1/2 (2.1 + 1.2/3)`

= `11/7 xx 1/2 xx (2.5)`

= `11/(7 xx 2) xx (2.5)`

= `11/14 xx (2.5)`

= `27.5/14`

= 1.964 m³

Remaining height of cylinder in contact with liquid (H') = 2.1 – 0.7 = 1.4 m

`l = sqrt(0.5^2 + 1.2^2)`

= `sqrt(0.25 + 1.44)`

= `sqrt(0.69)`

= `sqrt(169/100)`

= `13/10`

= 13

Required surface area = 2πrH' + πrl

= πr(2H' + l)

= `22/7 xx 1/2(2 xx 1.4 + 1.3)`

= `11/7(2.8 + 1.3)`

= `(11 xx 4.1)/7`

= `45.1/7`

= 6.44 m² (approx.)