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 PP is to be determined.

Studdy Solution

STEP 1

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

STEP 2

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

STEP 3

Calculate the distance d d from the light source to point P P using the Pythagorean theorem. The vertical distance from the light source to the bench is 2 2 meters, and the horizontal distance to point P P is 1.8 1.8 meters.
d=(2 m)2+(1.8 m)2 d = \sqrt{(2 \text{ m})^2 + (1.8 \text{ m})^2}
Calculate:
d=4+3.24 d = \sqrt{4 + 3.24} d=7.24 d = \sqrt{7.24} d2.69 m d \approx 2.69 \text{ m}

STEP 4

Use the inverse square law to find the illuminance E E at point P P . The illuminance is given by:
E=Id2 E = \frac{I}{d^2}
where I=800 cd I = 800 \text{ cd} and d2.69 m d \approx 2.69 \text{ m} .
Substitute the values:
E=800 cd(2.69 m)2 E = \frac{800 \text{ cd}}{(2.69 \text{ m})^2}
Calculate:
E=8007.24 E = \frac{800}{7.24} E110.5 lux E \approx 110.5 \text{ lux}
The illuminance at point P P is approximately:
110.5 lux \boxed{110.5 \text{ lux}}

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