Question

two right circular cones of equal volumes have their heights in ratio 1:2 then prove that their radii are in √2:1...plz answer Quickly

Answer 1

let the height n radius of the 2 cones be h1 h2 n r1 r2 respectively....
acc. to the ques...
h1/h2 = 1/2
h2=2 h1
now as volumes are equal
pi r1^2 h1= pi r2^2 h2
r1^2/r2^2=h1/h2
r1^2/r2^2=1/2
r1/r2=1/ \/2

Answer 2

Given

• two right circular cones
• equal volumes
• heights in ratio 1:2

To prove

• Their radii are in √2:1

Solution

we are provided with two right circular codes having equal volume and the ratio of their heights 1:2 and are asked to prove that the radius are in the ratio √2:1

volume of the cone is given by,

V = 1/3 πr^2 h

volume of both the cones are same therefore,

1/3 π (r1)^2 (h1) = 1/3 π(r2)^2 (h2). [where r1 and r2 are the radius of first and second cone respectively similarly h1 and h2 are heights of the cones.]

or, (r1)^2 (h1) = (r2)^2 (h2)

or, (r1/r2)^2 = (h2/h1)

or, (r1/r2)^2 = 2 /1. ( from given)

or, r1/r2 = √2/1

The radii of the cone are in the ratio √2:1