The volume of ice-cream in the cone is half the volume of the cone. The cone has a 3cm radius and 6cm height. What is the depth of the ice cream, correct to two decimal places?

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-measurements is in cm

1 Answer
Jan 28, 2018

h=4.76h=4.76 cmcm

Explanation:

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V_"Cone"=piR^2H/3VCone=πR2H3

R=3R=3 cmcm

H=6H=6

V_"Cone"=pi(3)^2(6/3)=18piVCone=π(3)2(63)=18π cm^3cm3

V_"Ice Cream"=(18pi)/2=9piVIce Cream=18π2=9π cm^3cm3

rr is the radius of the Ice Cream.

hh is the height (depth) of the Ice Cream.

The right triangles formed by the height of the cone and the two radii RR and rr are similar triangles by "AA"AA theorem. Therefore, the ratio of their corresponding sides are the same:

h/6=r/3h6=r3

6r=3h6r=3h

r=(3h)/6=h/2r=3h6=h2

V_"Ice Cream"=pir^2h/3VIce Cream=πr2h3

V_"Ice Cream"=pi(h/2)^2(h/3)VIce Cream=π(h2)2(h3)

9pi=h^3/12pi9π=h312π

9=h^3/129=h312

h^3=108h3=108

h=root(3)108=4.76h=3108=4.76 cmcm