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?

To find the size of the lamp needed to provide the same lighting at a distance of 12.0 ft, we can use the inverse square law of illumination.

The inverse square law states that the intensity of illumination is inversely proportional to the square of the distance from the light source. Mathematically, this can be expressed as:

I₁/I₂ = (d₂²/d₁²),

where I₁ is the initial intensity of illumination, I₂ is the desired intensity of illumination, d₁ is the initial distance, and d₂ is the desired distance.

In this case, the initial distance (d₁) is 8.00 ft and the desired distance (d₂) is 12.0 ft. We can assume that the initial intensity (I₁) is provided by a 74.0 W lamp, and we need to find the size of the lamp needed to provide the same lighting, which we'll call I₂.

First, let's calculate the initial intensity (I₁) using the initial distance (d₁) and the given lamp size of 74.0 W:

I₁ = (lamp size) / (distance)²
I₁ = 74.0 W / (8.00 ft)²

Next, let's solve for I₂ using the inverse square law:

I₁/I₂ = (d₂²/d₁²)
I₂ = I₁ * (d₁²/d₂²)

Finally, we can substitute the values and calculate I₂:

I₂ = (74.0 W / (8.00 ft)²) * ((8.00 ft)² / (12.0 ft)²)
I₂ = (74.0 W / 64.0 ft²) * (64.0 ft² / 144.0 ft²)
I₂ = 74.0 W * 64.0 ft² / (64.0 ft² * 144.0 ft²)
I₂ = 74.0 W / 144.0 ft²

Therefore, the size of the lamp needed to provide the same lighting at a distance of 12.0 ft is 74.0 W divided by 144.0 ft², or approximately 0.514 W/ft².