The illumination produced by a light source is inversely proportional to the square of the distance to the source. If a light source 5m from an object illuminates with 70 candela, what illumination would be produce if the object were mover to 12 m away?
To solve this problem, we need to use the inverse square law of illumination. According to the inverse square law, the illumination (I) produced by a light source is inversely proportional to the square of the distance (d) to the source. Mathematically, we can write this relationship as:
I α 1/d²
Where α is the symbol for proportional to.
Now, let's use this information to solve the problem.
First, we need to find the constant of proportionality in the equation. We can do this by using the given values:
I₁ = 70 candela (illumination when the object is 5m away)
d₁ = 5m (distance when the illumination is 70 candela)
Substituting these values into the equation, we get:
70 α 1/5²
Now, we can find the constant of proportionality by dividing both sides of the equation:
70 / (1/5²) = 70 * 25 = 1750
So, the equation becomes:
I α 1750
Next, we can use this equation to find the illumination (I₂) when the object is 12m away. We know that the constant of proportionality remains the same, so we can write:
I₁ / I₂ = d₂² / d₁²
Substituting the known values, we get:
70 / I₂ = 12² / 5²
Simplifying this equation gives:
70 / I₂ = 144 / 25
Cross-multiplying, we have:
144 * I₂ = 70 * 25
Now, solve for I₂:
I₂ = (70 * 25) / 144
I₂ ≈ 12.15 candela
Therefore, if the object is moved 12m away from the light source, the illumination produced would be approximately 12.15 candela.