Posted by **sara** on Monday, May 7, 2012 at 4:39am.

A light is placed 5 ft. from the center of a globe 3 ft. in diameter. Find the area of the illuminated portion.

thank you!

- math (solid mensuration) -
**Reiny**, Monday, May 7, 2012 at 9:02am
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

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