# For the given information develop equation which relates intensity of illumination I inversely to square of distance d

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1, Problem 135RE

a.

To determine

## To calculate:For the given information develop equation which relates intensity of illumination I inversely to square of distance d

Expert Solution

The required equation is I=kd2

### Explanation of Solution

Given information:

The Intensity of illumination I from a given light is inversely proportional to square of distance d from the given light.

Formula used:

For 2 variables say, x and y , the statement x is directly proportional to y can be written as:

xαy

Which can be written as:

x=ky

Where k denotes the proportionality constant.

Similarly the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Calculation:

AsIntensity of illumination I from a given light is inversely proportional to square of distance d from the given light.

Recall, For 2 variables say, x and y , the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Hence, this variation can be expressed as follows:

Iα1d2I=kd2

Where k denotes the proportionality constant.

Hence, the required equation is I=kd2

b.

To determine

### To calculate:The constant of proportionality k if the lamp intensity is 1000 candles and the distance is 8 meters.

Expert Solution

The constant of proportionality k if the lamp intensity is 1000 candles and the distance is 8 meters is k=64000 candles per m2

### Explanation of Solution

Given information:

The equation found is I=kd2 and the lamp intensity is 1000 candles while the distance from the light is 8 meters

Formula used:

For 2 variables say, x and y , the statement x is directly proportional to y can be written as:

xαy

Which can be written as:

x=ky

Where k denotes the proportionality constant.

Similarly the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Calculation:

AsIntensity of illumination I from a given light is inversely proportional to square of distance d from the given light.

Form a. the required equation is I=kd2 (1)

It is given that the intensity of illumination I=1000 and distance from the light d=8 meters.

Put these values in (1) to get:

1000=k82k=64*1000k=64000

Hence, the constant of proportionality is k=64000 candles per m2

c.

To determine

### To calculate:The intensity of lamp when the distance from the light is 20 meters

Expert Solution

The intensity of lamp when the distance from the light is 20 meters is I=160 candles.

### Explanation of Solution

Given information:

The equation found is I=kd2 and the constant of proportionality is k=64000 candles per m2 . Also distance from the light is d=20 meters

Formula used:

For 2 variables say, x and y , the statement x is directly proportional to y can be written as:

xαy

Which can be written as:

x=ky

Where k denotes the proportionality constant.

Similarly the statement x is inversely proportional to y can be interpreted as:

xα1y

Which can be written as:

x=k(1y)

Where k denotes the proportionality constant.

Calculation:

AsIntensity of illumination I from a given light is inversely proportional to square of distance d from the given light.

Form a. the required equation is I=kd2 (1)

From b. the constant of proportionality is k=64000 candles per m2 (2)

Also, it is given distance from the light is d=20 meters (3)

Put (2) and (3) in (1) to get:

I=64000202I=64000400I=160

Hence, the intensity of lamp when the distance from the light is 20 meters is I=160 candles.

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