math (solid mensuration)
posted by sara on .
A light is placed 5 ft. from the center of a globe 3 ft. in diameter. Find the area of the illuminated portion.
make a side view of the event
draw a line from the light to the centre of the sphere,
draw a line to the sphere representing your line of sight, this would be a tangent to the sphere
It is easy to see, that this tangent would be 4 ft
(the 3-4-5 right-angled triangle)
the angle Ø at the centre of that triangle is sin^-1 (4/5) = appr 53.13°
so we need the surface area of the "cap".
let x be the distance from centre to base of the cap
cos 53.13 = x/3
.6 = x/3
x = 1.8
so the height , h, of the cap is 3-1.8 = 1.2
The curved surface area of the spherical cap is 2πrh
where r is the radius of the sphere and h is the height of the cap
SA = 2π(3)(1.2) = 7.2π square feet