# Calculus

posted by .

A light is suspended at a height h above the floor. The illumination at the point P is inversely proportional to the square of the distance from the point P to the light and directly proportional to the cosine of the angle θ. How far from the floor should the light be to maximize the illumination at the point P? (Let d = 6 ft.)

• Calculus -

I guess d is from below the light to a point on the floor?
If so:

I = k cos T / (h^2+d^2)
where cos T = h/(h^2+d^2)^.5
so
I = k h/(h^2+d^2)^1.5
dI/dh = k [ (h^2+d^2)^1.5 -h(1.5)(h^2+d^2)^.5 (2h) ] / (h^2+d^2)^3
= 0 for extreme
(h^2+d^2)^1.5 = 3 h^2 (h^2+d^2)^.5
(h^2+d^2) = 3 h^2
d^2 = 2 h^2
d = h sqrt 2
h = 6/sqrt 2 = 4.24 ft

## Similar Questions

1. ### math

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the …
2. ### calculus

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the …
3. ### Calculus

"The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two streetlights are 40m apart and one is twice as strong …
4. ### Math

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one four times as strong as the …
5. ### Algebra II

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 …
6. ### Algebra II

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 …
7. ### Algebra II

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 …
8. ### Math

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 …
9. ### MATH 100

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one 10 times as strong as the other, …
10. ### Math

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one four times as strong as the …

More Similar Questions