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Q) A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-cream. The whole ice-cream is to be divided into 10 children in equal ice-cream cones, with conical base surmounted by hemispherical top. If the height of conical portion is twice the diameter of base, find the diameter of conical part of ice-cream cones.

Ans:

Step 1: Let’s draw a diagram to better understand the question:

A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-creamIn the above diagram, the ice-cream of cylindrical tub (diameter 12 cm, height 15 cm) is converted into 10 cones of (diameter d cm, height 2 d cm).

We consider that the ice-cream cone is fully filled by the ice-cream and there is no empty space in the cone.

Step 2: Let’s first start with volume of ice-cream in the tub.

we have diameter of cylinder, Dcyl  = 12 cm

∴ radius of cylinder, Rcyl  = \frac{12}{2} = 6 cm

given the height of cylinder, Hcyl  = 15 cm

We know that the volume of a cylinder is given by: π r2 h

∴ the volume of ice-cream in the cylindrical tub, Vcyl  = π rcyl 2  h cyl

= π (6)2 (15) = 540 π

Step 3: We will now calculate ice-cream in the cone:

Part 1: Since we considered the diameter of cone, d cm

Therefore the radius of the cone, rC  = \frac{d}{2} cm

and the height of the cone, hC = 2 d cm (given that the height of the cone is twice the diameter)

Since, we know that the volume of the cone, VC = \frac{1}{3} \pi (\frac{d}{2})^2 (2 d)

= \frac{d^3}{6} \pi

Part 2: Icecream outside the cone is hemispherical

Its radius, rH = \frac{d}{2} cm (radius of the hemisphere is same as of conical base)

Since, we know that the volume of a hemisphere, VH = \frac{2}{3} \pi (r ^3)

= \frac{2}{3} \pi (\frac{d}{2})^3 = \frac{d^3}{12} \pi

Part 3: The volume of conical ice-cream with hemispherical top = VC  + VH 

= \frac{d^3}{6} \pi + \frac{d^3}{12} \pi

= \frac{d^3}{4} \pi

Step 4: Volume of ice cream in 10 such cones = 10 x \frac{d^3}{4} \pi

= \frac{5 d^3}{2} \pi

Step 5: Since spherical tub’s ice-cream is being converted into 10 conical ice-creams with hemispherical tops

∴ Volume of spherical tub’s ice-cream = ice-cream in 10 cones with hemispherical tops

∴ 540 \pi = \frac{5 d^3}{2} \pi

∴ 540 x 2 = 5 d 3

∴ d 3 = \frac {1080}{5}

∴ d 3 = 216 = (6)3

∴ d = 6 cm

Therefore, the diameter of conical part of ice-cream cones is 6 cm.

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