The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 74.0 w lamp 8.00 ft from the desk, what size lamp will be needed to provide the same lighting at a distance of 12.0 ft?

112

To find the size of the lamp needed to provide the same lighting at a different distance, we can use the principle of inverse square law for illumination. The formula for the intensity of illumination is:

I = k * (1/d^2)

Where:
I is the intensity of illumination,
k is the proportionality constant,
d is the distance from the light source.

We can set up a proportion using the given information:

I1 = k * (1/8^2) --> Equation 1
I2 = k * (1/12^2) --> Equation 2

Since we want to find the size of the lamp needed for the same lighting, we can set I1 = I2:

I1 = I2

k * (1/8^2) = k * (1/12^2)

Simplifying the equation:

1/8^2 = 1/12^2

1/64 = 1/144

Cross-multiplying:

144 = 64

This equation is not valid, so the assumption that the intensity is directly proportional to the inverse square of the distance is not accurate. In real-world scenarios, other factors like light diffraction, absorption, reflection, and scattering also affect the intensity of illumination. Therefore, we cannot determine the exact size of the lamp needed based solely on this information.