Calculus, 8th Edition,stewart; page 36 problem 29

Question

Many physical quantities are connected by inverse square laws, that is, by power functions of the form f (x) = k x^(-2). In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lump and you  are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light.

Answer 1

Step:1

Inverse square law :

An object by a light source is inversely proportional to square of the distance from the source.

Step:2

In the dark room the person is trying to read a book.

Consider the distance between the person and light source is .

From the inverse square law .

Light is too dim, so the person moves to half distance to the lamp.

Then the distance between the person and the light source is .

From the inverse square law,

Step:3

Find brightness of the light if he moves to half distance to the lamp.

Ratio between  and  gives the brightness of the light if he moves to half distance to the lamp.

The light is  times brighter when the person moves to half distance to the lamp.

Solution:

The light is times brighter when the person moves to half distance to the lamp.