Question

In the given figure, ABCDE represents the bowl of a concrete mixer. ABDE can be a part of a cone FDB as shown below where base radius OD = 30 cm, OP = 20 cm and PF = 1 m.

1. Calculate the value of PE using similarity of triangles.
2. Calculate the volume of the part with cross-section ABDE.
3. Calculate the volume of the whole concrete mixer to the nearest litre.

Answer 1

(i) F is the apex of the cone FDB. P and O lie on the axis with PF = 1 m = 100 cm and OP = 20 cm.

∴ FO = FP + PO = 100 + 20 = 120 cm.

In △FPE and △FOD :

∠PFE = ∠OFD (common angle)

∠FPE = ∠FOD (each = 90°, both perpendicular to the axis)

∴ △FPE ∼ △FOD (by AA axiom)

We know that,

In similar triangles corresponding sides are proportional.

\[\Rightarrow \frac{PE}{OD} = \frac{FP}{FO}\]

\[\Rightarrow \frac{PE}{30} = \frac{100}{120}\]

\[\Rightarrow PE = \frac{100}{120} \times 30 = 25\ \text{cm}.\]

Hence, PE = 25 cm.

(ii) From figure,

For cone FBD:

Radius (R) = 30 cm

Height (H) = 120 cm

For cone FAE:

Radius (r) = 25 cm

Height (h) = 100 cm

Volume of part with cross section ABDE (V) = Volume of cone FBD − Volume of cone FAE

\[\Rightarrow V = \frac{1}{3} \times \frac{22}{7} \times 30^2 \times 120 - \frac{1}{3} \times \frac{22}{7} \times 25^2 \times 100\]

\[\Rightarrow V = \frac{1}{3} \times \frac{22}{7} \times 900 \times 120 - \frac{1}{3} \times \frac{22}{7} \times 625 \times 100\]

\[\Rightarrow V = \frac{1}{3} \times \frac{22}{7} \times (108000 - 62500)\]

\[\Rightarrow V = \frac{1}{3} \times \frac{22}{7} \times 45500\]

\[\Rightarrow V = \frac{22}{3} \times 6500\]

\[\Rightarrow V = \frac{143000}{3} = 47666\frac{2}{3} \approx 47666.67\ \text{cm}^3\]

(iii) From figure,

For hemisphere BCD,

Radius (R) = 30 cm

Volume of hemisphere \[BCD = \frac{2}{3}\, \pi R^3\]

Volume of hemisphere \[BCD = \frac{2}{3} \times \frac{22}{7} \times (30)^3\]

\[{} = \frac{2}{3} \times \frac{22}{7} \times 27000\]

\[{} = \frac{22}{7} \times 18000\]

\[{} = \frac{396000}{7} = 56571\frac{3}{7} \approx 56571.43\ \text{cm}^3\]

Volume of whole concrete mixer = Volume of part with cross section ABDE + Volume of hemisphere BCD

= 47666.7 + 56571.4

= 104238.1 cm³

= 104238.1 × 0.001 liters

= 104.238 liters.

Hence, the volume of the whole concrete mixer ≈ 104 litres.