Cylinder, Cone & Sphere (Surface Area & Volume) — Case-Study Based 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 radius OD = 30 cm, OP = 20 cm and PF = 1 m.

(i) Calculate the value of PE using similarity of triangles.

(ii) Calculate the volume of the part with cross-section ABDE.

(iii) Calculate the volume of the whole concrete mixer to the nearest litre.

Answer

(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.

$\Rightarrow \dfrac{PE}{OD} = \dfrac{FP}{FO} \\[1em] \Rightarrow \dfrac{PE}{30} = \dfrac{100}{120} \\[1em] \Rightarrow PE = \dfrac{100}{120} \times 30 = 25 \text{ cm}.$

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

$\Rightarrow V = \dfrac{1}{3} \times \dfrac{22}{7} \times 30^2 \times 120 - \dfrac{1}{3} \times \dfrac{22}{7} \times 25^2 \times 100 \\[1em] \Rightarrow V = \dfrac{1}{3} \times \dfrac{22}{7} \times 900 \times 120 - \dfrac{1}{3} \times \dfrac{22}{7} \times 625 \times 100 \\[1em] \Rightarrow V = \dfrac{1}{3} \times \dfrac{22}{7} \times \left(108000 - 62500\right) \\[1em] \Rightarrow V = \dfrac{1}{3} \times \dfrac{22}{7} \times 45500 \\[1em] \Rightarrow V = \dfrac{22}{3} \times 6500 \\[1em] \Rightarrow V = \dfrac{143000}{3} = 47666\dfrac{2}{3} \approx 47666.67 \text{ cm}^3$

Hence, volume of part with cross section ABDE = 47666.7 cm³.

(iii) From figure,

For hemisphere BCD,

Radius (R) = 30 cm

Volume of hemisphere BCD = $\dfrac{2}{3}πR^3$

$\text{Volume of hemisphere BCD} = \dfrac{2}{3} \times \dfrac{22}{7} \times (30)^3 \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times 27000 \\[1em] = \dfrac{22}{7} \times 18000 \\[1em] = \dfrac{396000}{7} = 56571\dfrac{3}{7} \approx 56571.43 \text{ cm}^3$

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.