find height of the cone by using volume of cone formula and this question from RS Aggarwal, Volume and Surface Area of Solid, Exercise 17A, problem number 5, Page number 786

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# If the volumes of two cones are in the ratio of 1:4 and their diameters are in the ratio of 4:5, then find the ratio of their heights.

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Volume of cone : (πr²h)/3

Where r= radius of cone and h = height of cone.

Let V1 be the volume of one cone and V2 the other.

Then V1=(π(r1)² h1)/3

V2=(π [r2]² h2)/3

V1:V2=1:4

V2=4V1. This leads to

(π/3)(r2)² h2=4*(π/3)(r1)² h1

Simplifying the equation ,we have

(r2)² *h2=4*(r1)² *h1…1

Also the diameters are in the ratio 4:5.

So r1:r2=4:5

4r2=5r1

r2=(5/4)r1

Substituting these in equation 1

{(5/4)r1}² * h2=4*(r1)² * h1

(25/16) * h2 = h1 * 4

25 h2 = 4 * 16h1

25 h2 = 64h1

h1:h2=25:64

The heights are in the ratio 25:64