Exercise 1.2.23

Question

The illumination of an object by a light source is related to the distance from the source by an inverse square law. Suppose that after dark you are sitting in a room with just one lamp, trying to read a book. The light is too dim, so you move your chair halfway to the lamp. How much brighter is the light?

Khagan Gulusoy · Accepted Answer

Since the illumination of an object by a light source is related to the distance from the source by an inverse square law, we have:
$$\forall x \in [0,+\infty): \quad I(x)=\frac{C}{x^2},$$
where $I$ is the illumination of an object by a light source, $x$ is the distance from the source, and $C$ is some positive real number. Let $x_1$ denote the original distance between the lamp and the source of light. When we move the chair closer to the lamp, the distance between the chair and the lamp reduces by half, which means it becomes $x_2:= \dfrac{x_1}{2}$. Therefore, after moving the chair, the illumination becomes
$$I(x_2)= I\left(\frac{x_1}{2}\right) = \frac{C}{\left(\frac{x_1}{2}\right)^2}=\frac{4C}{x_1^2} = 4 \cdot I(x_1),$$
%where $I_2$ is the illumination of the object after the movement described in the exercise. Thus, we have:
%$$I_2=4I.$$
In other words, the current light is now four times brighter than the initial illumination level.

Exercise 1.2.23

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Exercise 1.2.23

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