Problem 54 - College Algebra

Question

The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?

Step-by-Step Solution

Answer

The illumination on the desk decreases to one quarter of the original intensity when the lamp is raised from 15 inches to 30 inches.

Step 1: Understand Inverse Square Law

The Inverse Square Law can be represented as I = k/d^2, where I represents the illumination, d is the distance from the light source, and k is the constant of proportionality. It signifies that the illumination decreases with the increase in the square of the distance.

Step 2: Express given conditions in terms of the law

As per the question, the lamp is initially 15 inches over the desk and is then raised to 30 inches. Let's denote the initial illumination when the lamp was 15 inches above the desk as I1 and the new illumination when the lamp is 30 inches high as I2.

Step 3: Apply the inverse square law

Since the illumination from a light source varies inversely as the square of the distance from the light source, we can formulate these equations: I1 = k/(15)^2 & I2 = k/(30)^2.

Step 4: Compare the two conditions

From the equations in step 3, we get I1/I2 = (k/(15)^2) / (k/(30)^2) = (30/15)^2 = 2^2 = 4. Therefore, the initial illumination was four times the final illumination.

Step 5: Conclusion

Therefore, when the lamp is raised from 15 inches to 30 inches, the illumination on the desk reduces to one quarter of the original illumination. This is due to the inverse square law which states that the illumination varies inversely as the square of the distance from the light source.