The illumination a surface receives from a light source varies inversely with the square of the distance. A table lamp is 3ft from a certain work area. How far is another work area if it receives only 1/3 as much illumination as the first work area?
(W/ solution please, tnx...)
"The illumination a surface receives from a light source varies inversely with the square of the distance."
-----> I = k(1/d^2) , where I is the illumination, and d is the distance
or:
I1/I2 = d2^2/d1^2
I/(1/3)I) = 3^2 / d1^2
3 = 9/d^2
3d^2= 0
d^2 = 3
d = √3
Let's call the illumination of the first work area I1 and the illumination of the second work area I2. We can set up the equation:
I1 = k/d1^2
where k is a constant and d1 is the distance from the lamp to the first work area.
We are given that the first work area is 3ft from the lamp, so we have:
I1 = k/3^2
Now, we want to find the distance, d2, from the lamp to the second work area. We are also given that the second work area receives only 1/3 as much illumination, so we have:
I2 = 1/3 * I1
Substituting the value of I1, we can write:
1/3 * k/3^2 = k/d2^2
Simplifying, we have:
1/9 = k/d2^2
Now, we can solve for d2:
d2^2 = k/(1/9)
d2^2 = 9k
Taking the square root of both sides, we get:
d2 = √(9k)
So, the distance from the lamp to the second work area is √(9k).
To solve this problem, we can use the inverse square law for illumination, which states that the illumination on a surface varies inversely with the square of the distance.
Let's denote the illumination on the first work area as I1 and the distance from the lamp to the first work area as d1. Similarly, let's denote the illumination on the second work area as I2 and the distance from the lamp to the second work area as d2.
From the given information, we know that the illumination on the second work area (I2) is 1/3 of the illumination on the first work area (I1). Mathematically, we can express this as:
I2 = (1/3) * I1
We also know that the distance from the lamp to the first work area (d1) is 3 ft. We need to find the distance from the lamp to the second work area (d2).
According to the inverse square law, the illumination is inversely proportional to the square of the distance. This can be written as:
I2 / I1 = (d1^2) / (d2^2)
Substitute the given values into the equation:
(1/3) * I1 / I1 = (3^2) / (d2^2)
Simplify the equation:
1/3 = 9 / (d2^2)
Cross-multiply:
d2^2 = 9 * 3
d2^2 = 27
Take the square root of both sides to find d2:
d2 = √27
Simplify:
d2 ≈ 5.196 ft
Therefore, the second work area is approximately 5.196 ft from the lamp.