A small light source located 1 m in front of a 1 m^2 opening illuminates a wall behind. If the wall is 1 m behind the opening (2 m from the light source), the illuminated area covers 4 m^2. How many square meters will be illuminated if the wall is 3 m from the light source?
If it is 3m from the light source I think it will illuminate 9m^2. I know if it is 5m from the source it illuminates 25m^2 and also if it is 10m from the source it illuminates 100m^2 so it makes sense that 3m would illuminate 9m^2.
To calculate the area that will be illuminated when the wall is 3 m from the light source, we can use the inverse square law. This law states that the intensity of light is inversely proportional to the square of the distance from the light source.
Given that the illuminated area is 4 m^2 when the wall is 2 m from the light source, we can set up the following ratio:
(Intensity at 1 m) / (Intensity at 2 m) = (Area at 2 m) / (Area at 1 m)
Let's denote the intensity at 1 m as I1 and the intensity at 2 m as I2. We know that the area at 1 m is 1 m^2, and the area at 2 m is 4 m^2. Plugging in the values, we have:
I1 / I2 = 4 / 1
We can rearrange the equation to solve for I1:
I1 = (4 / 1) * I2
Now, if we want to find the area that will be illuminated when the wall is 3 m from the light source, we can set up another ratio:
(Intensity at 1 m) / (Intensity at 3 m) = (Area at 3 m) / (Area at 1 m)
Let's denote the area at 3 m as A3. Plugging in the values, we have:
I1 / I3 = A3 / 1
Since we already know the value of I1 as (4 / 1) * I2, we can substitute it into the equation:
(4 / 1) * I2 / I3 = A3 / 1
We know that I2 is the intensity at 2 m, which we assume remains constant regardless of the distance of the wall from the light source. Therefore, we can simplify the equation to:
4 / I3 = A3 / 1
Or, rearranging the equation to solve for A3:
A3 = (4 / I3) * 1
Now we need to determine the value of I3, the intensity at 3 m from the light source. Using the inverse square law, we know that:
I1 / I2 = (Distance2 / Distance1)^2
Plugging in the values, we have:
I1 / I2 = (2 / 1)^2
Simplifying, we get:
I1 / I2 = 4 / 1
Therefore, I1 = 4 * I2.
Substituting the value of I1 into the equation, we have:
A3 = (4 / I3) * 1 = (4 / (4 * I2)) * 1 = 1 / I2
Hence, the area that will be illuminated when the wall is 3 m from the light source is given by 1 / I2 square meters.
To find out how many square meters will be illuminated if the wall is 3 m from the light source, we can use the inverse square law for light intensity.
The inverse square law states that the intensity of light decreases as the square of the distance between the source and the surface being illuminated. Mathematically, it can be expressed as:
I = k / d^2
Where:
- I is the intensity of light
- k is a constant
- d is the distance between the light source and the surface
Let's use the information given in the question to solve for the constant k.
Given that the light is illuminating a 1 m^2 opening located at a distance of 1 m from the light source, we can use the inverse square law to determine the light intensity at that distance:
I1 = k / (1^2)
I1 = k
We are also given that the light illuminates a 4 m^2 area when the wall is located 2 m from the light source:
I2 = k / (2^2)
I2 = k / 4
Now, let's find the value of k by equating I1 and I2:
k = k / 4
4k = k
3k = 0
Since we obtain an equation with no valid solution (3k=0), the given information seems to be inconsistent.
Therefore, we cannot accurately determine how many square meters will be illuminated when the wall is 3 m from the light source based on the provided information.