The height of a cone is 20cm and the radius is 8cm from the top of 3cm cutted from the cone find the ratio of their volumes
Answers
Answer:
H = 20 cm = height of the right circular cone.
R cm = radius of the base of the cone
V = Volume of the Cone = 1/3 * π R² * H
Let the radius of the base of the small cone = r cm
h = height of the small cone.
v = volume of small cone = 1/3 π * r² * h
From the similar triangles principles,
r / h = R / H
r = R h / H
given V = 8 v
=> 1/3 π R² H = 8 * 1/3 π r² h
=> R² H = 8 * r² h
=> R² H = 8 * (R² h² / H²) * h
=> H³ = 8 h³
=> h = H/2
=> h = 20 cm / 2 = 10 cm
============================
another way:
When a small cone is cut off from the top of the cone, the ratio of the radii of the bases is equal to the ratio of the heights.
R / r = H / h = k (let us say)
R = k r and H = k h
Ratio of volumes = (π/3 R² H) / (π/3 r² h) = 8 given
=> ( k² r² k h ) / ( r² h ) = 8
=> k³ = 8
k = ∛8 = 2
=> H = 2 h and R = 2 r
Hence, the height of the small cone = H/2 = 10 cm.
Answer:
Volume of frustum =
3
1
π(R +Rr)×h
=
3
1
×3((0.1)
2
+(0.01)
2
+(0.1×0.01))×3
=3(0.01+0.0001+0.001)
=3×0.0111=0.0333cm
3