Intensity of Light. The intensity I of light from a light bulb varies inversely as the square of the distance d from the bulb. Suppose that I is
90
W
/
m
2
90W/m2 (watts per square meter) when the distance is 5 m. How much farther would it be to a point where the intensity is
40
W
/
m
2
40W/m2?
I = k/d^2
so, Id^2 = k is constant
So, find d such that
40d^2 = 90*5^2
and subtract 5
To find out the distance to a point where the intensity is 40W/m^2, we'll use the inverse square law formula:
I1 * d1^2 = I2 * d2^2
where:
I1 = initial intensity (90W/m^2)
d1 = initial distance (5m)
I2 = final intensity (40W/m^2)
d2 = final distance (unknown)
First, let's plug in the given values into the formula:
90 * 5^2 = 40 * d2^2
Simplifying further:
450 = 40 * d2^2
Now, divide both sides of the equation by 40:
450/40 = d2^2
11.25 = d2^2
To find d2, take the square root of both sides:
√(11.25) = √(d2^2)
d2 ≈ 3.35
Therefore, it would be approximately 3.35 meters farther to a point where the intensity is 40W/m^2.
To find out how much farther it would be to a point where the intensity is 40W/m2, we need to use the inverse square law equation for intensity of light:
I = k/d^2
where I represents the intensity, k is the constant of variation, and d is the distance from the source of light.
We are given that the intensity is 90W/m2 when the distance is 5m. Plugging these values into the equation, we get:
90 = k/5^2
90 = k/25
To solve for k, we can multiply both sides of the equation by 25:
90 * 25 = k
2250 = k
Now that we have the value of k, we can use it to find the distance (d) where the intensity is 40W/m2. Plugging in the values into the inverse square law equation:
40 = 2250/d^2
To solve for d, we need to isolate d^2 on one side of the equation. We can do this by dividing both sides of the equation by 40:
40/40 = 2250/d^2 / 40
1 = 2250/d^2 / 40
1 = 2250/d^2 * 1/40
1 = 2250/(40*d^2)
Now we can solve for d by taking the reciprocal of both sides:
1/1 = 40*d^2 / 2250
1 = 40*d^2 / 2250
To get d^2 alone, we can multiply both sides of the equation by 2250:
2250 = 40*d^2
Now divide both sides by 40:
2250/40 = d^2
56.25 = d^2
Finally, take the square root of both sides:
√56.25 = √d^2
d ≈ 7.50 m
Therefore, it would be approximately 7.50 meters farther to a point where the intensity is 40W/m2.