Volumes of ball parts and the surface of a sphere
Answers are for parents and teachers.
1
A ball has volume 288π. Find its radius and surface.
(4/3)R³ = 288, R³ = 216, so R = 6; S = 4π · 36 = 144π.
2
Find the area of a cap of height 5 on a sphere with R = 12.
2π · 12 · 5 = 120π.
3
A metal ball of radius 10 cm is melted and cast into 8 equal small balls. Find the radius of a small ball and how many times their total surface exceeds that of the big ball.
Volumes: 8r³ = 1000, so r = 5 cm. The total surface is 8 · 4π · 25 = 800π cm² versus 400π cm² for the big ball: twice as much.
4
Find the volume of a spherical segment with R = 6 and h = 4.
V = π · 16 · (18 − 4)/3 = 224π/3.
5
Derive S = 4πR² from V = (4/3)πR³ and V = ⅓ S R.
S = 3V/R = 3 · (4/3)πR³/R = 4πR².