Question

A child consumed an ice-cream of inverted right-circular conical shape from the top and left only 12.5% of the cone for her mother. If the height of the ice-cream cone was 8 cm, what was the height of the remaining ice-cream cone ? 

(a) 2.5 cm 

(b) 3.0 cm 

(c) 3.5 cm 

(d) 4.0 cm

Answer 1

(d) 4

Let ADE be the remaining portion of the cone. Then, from similar Δs AFE and AGC 

\(\frac{AF}{AG}=\frac{FE}{GC}\) ⇒ FE = \(\frac{AF.GC}{AG}\)

⇒ r₁ = \(\frac{hr}{8}\)

where r₁ = radius of cone ADE, 

h = height of cone ADE and 

r = radius of filled cone ABC 

Now, volume of smaller cone 

ADE = \(\frac13πr^2_1h = \frac13π\big(\frac{hr}{8}\big)^2.h\)

= \(\frac13\timesπ\times\frac{h^3r^2}{64}\)

Volume of bigger cone ABC = \(\frac13πr^2\times8\)

Given, \(\frac13\timesπ\times\frac{h^3r^2}{64}=12.5\)% of \(\big(\frac83πr^2\big)\)

⇒ \(\fracπ3\times\frac{h^3r^2}{64}=\frac{125}{1000}\times\frac83πr^2\) ⇒ h³ = 64 ⇒ h = 4.