Problem 60 - 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

When the lamp is raised from 15 inches to 30 inches, the illumination on the desk will get quartered.

Step 1: Identify the relationship

In this kind of problem where the illumination varies inversely as the square of the distance from the light source, the mathematical representation can be \[ I = \frac{k}{d^2} \] where \(I\) represents the illumination, \(d\) is the distance from the source, and \(k\) is the constant of variation which remains consistent throughout the problem.

Step 2: Determine the relation between the original distance and the new distance

According to the problem, the lamp is raised from 15 inches to 30 inches. So, the distance is doubled.

Step 3: Apply the inversely proportional principle

As the illumination \(I\) is inversely proportional to the square of distance, when the distance is doubled, the illumination will get cut to a quarter because the new illumination \(I'\) would be \[ I' = \frac{k}{(2d)^2} = \frac{k}{4d^2} = \frac{I}{4}\]. So, the illumination is quartered.