# "A light is hung from rafter at a height h m above the floor , providing a circle of illumination. the illuminance E (in lumens/ squares meter or lm/`m^(2)` ) at ant point p on the floor varies...

"A light is hung from rafter at a height ** h **m above

*the floor , providing a circle of illumination.*

*the illuminance E (in lumens/ squares meter or lm/`m^(2)` ) at ant point p on the floor varies directly as the cosine of angle `theta` at which the rays leave the light source toward P, and inversely as the square of the height h of the light source, using k as the constant of variation, write expression for illuminance in term*

**k, h,**and `theta`. If the distance from the light source to point**P**is 13 m*write the expression in illuminance in terms of k and h alone.*

*for k = 15,600, what is the height of the light source, if E = 100lm/m^2?*

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### 1 Answer

Given, Illuminance E, at any point P varies directly as the cosine of angle `theta` at which the rays leave the light source toward P.

Mathematically, `E prop costheta`

Again, illuminance E, at any point P varies inversely as the square of the height h of the light source from the point P (refer to the attached diagram).

`E prop 1/h^2`

Combining these two relations,

`E prop costheta/h^2`

`rArr ` `E=(k*costheta)/h^2` --- (i) where, k is the constant of proportionality

This is the expression for illuminance in terms of k, h and theta.

Now, triangle PCL is a right triangle, `costheta=h/(PL)=h/13`

Therefore, `E= k*costheta`

`rArr E=(k* h)/13*1/h^2=k/(13h)` --- (ii)

This is the expression for illuminance in terms of k and h*.*

Given k=15600, E=100 lm/m^2, plugging in the values in equation (ii),

`100=15600/(13h)`

`rArr` `h=12 m`

**Therefore, for k = 15,600 and E =100 lm/m^2, the height of the light source, is 12 m.**

**Sources:**