The illumination provided by a car’s headlight varies inversely as the square of the distance from the headlight. If a headlight produces 15 footcandles at a distance of 20 feet, what will be the illumination at 40 feet? At 100 feet?
You are told that
I(d) = k/d^2
15 = k/400
k = 6000
I(40) = 6000/1600
I(100) = 6000/10000
THANK YOU!!
To solve this problem, we can use the inverse square law equation:
I = k/d^2
Where:
I = illumination (in footcandles)
k = constant of proportionality
d = distance from the headlight (in feet)
We are given that the illumination at a distance of 20 feet is 15 footcandles. So, we can substitute these values into the equation to find the value of k:
15 = k/20^2
15 = k/400
k = 15 * 400
k = 6000
Now that we have the value of k, we can use the equation to find the illumination at different distances.
For a distance of 40 feet:
I = 6000/40^2
I = 6000/1600
I = 3.75 footcandles
For a distance of 100 feet:
I = 6000/100^2
I = 6000/10000
I = 0.6 footcandles
Therefore, the illumination at 40 feet is 3.75 footcandles, and the illumination at 100 feet is 0.6 footcandles.