Math / Geometry

QuestionA 800 cd incandescent lamp is fixed at a height of 2 meters directly above a long bench, and the value of illuminance at point $P$ is to be determined.

Studdy Solution

STEP 1

1. The light source emits uniformly in all directions.
2. The illuminance at point $P$ is calculated using the inverse square law.
3. The angle $\theta$ and distance $d$ are related through trigonometry.

STEP 2

1. Calculate the distance $d$ from the light source to point $P$ .
2. Use the inverse square law to find the illuminance $E$ at point $P$ .

STEP 3

Calculate the distance $d$ from the light source to point $P$ using the Pythagorean theorem. The vertical distance from the light source to the bench is $2$ meters, and the horizontal distance to point $P$ is $1.8$ meters.
$d = \sqrt{(2 \text{ m})^2 + (1.8 \text{ m})^2}$
Calculate:
$d = \sqrt{4 + 3.24}$ $d = \sqrt{7.24}$ $d \approx 2.69 \text{ m}$

STEP 4

Use the inverse square law to find the illuminance $E$ at point $P$ . The illuminance is given by:
$E = \frac{I}{d^2}$
where $I = 800 \text{ cd}$ and $d \approx 2.69 \text{ m}$ .
Substitute the values:
$E = \frac{800 \text{ cd}}{(2.69 \text{ m})^2}$
Calculate:
$E = \frac{800}{7.24}$ $E \approx 110.5 \text{ lux}$
The illuminance at point $P$ is approximately:
$\boxed{110.5 \text{ lux}}$

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