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How to determine the length of a Shadow?

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We can determine the length of a shadow by determining two parameters first the angle of the light source, and second the distance between the object and the surface onto which the shadow is cast and it is necessary to determine position of object. The formula for calculating the length of shadow when angle is usually smaller than 10 degree is: length of Shadow = Distance from Object to Shadow Point × tan(Angle of Elevation) where “Distance from Object to Shadow Point” is the distance between the object and the point on the ground where the shadow falls. Depending on how the light source and the item are oriented, you might need to employ more complicated trigonometric functions like sine or cosine if the angle of elevation is greater or if you need greater precision.

What is Shadow?

A shadow of an object is created when it gets in the way of travelling light. When light rays come in contact with an opaque vertical object, it gives rise to a horizontal line of some length, which we call by the name of shadow. The angle of elevation is utilised to calculate the shadow length. It is the angle formed by the horizontal line (shadow) and the line of sight (horizon).

Shadow Length Formula

In the above figure, when light rays fall on the object AB, a horizontal line or its shadow OB is created. The line OA is called the line of sight as the angle of elevation θ is formed at the point O. The shadow length formula is equal to the height of the object whose shadow is formed divided by the tangent of the angle of elevation.

s = h/tan θ

where, 

s is the shadow length,

h is the height of object,

θ is the angle of elevation.

Derivation

Consider an object AB of height h such that its shadow of length s is formed and angle of elevation is θ.

The above figure of the object AB depicts a right triangle in which ∠ABO = 90o.

Now we know that in trigonometry, the tangent ratio for an angle is equal to opposite side to that angle divided by its adjacent side.

So, for the angle of elevation θ, we have

=> tan θ = AB/OB

Putting AB = h and OB = s, we get

=> tan θ = h/s

=> s/h = 1/tan θ

=> s = h/tan θ

 This derives the formula for shadow length of an object.

Sample Problems

Problem 1. Find the length of the shadow of an object at a height of 5 m if the angle of elevation is 45o.

Solution:

We have,

h = 5

θ = 45o

Using the formula for shadow length, we have

s = h/tan θ

= 5/tan 45o

= 5/1

= 5 m

Problem 2. Find the length of the shadow of an object at a height of 7 m if the angle of elevation is 60o.

Solution:

We have,

h = 7

θ = 60o

Using the formula for shadow length, we have

s = h/tan θ

= 7/tan 60o

= 7/√3

= 4.04 m

Problem 3. Find the length of the shadow of an object at a height of 12 m if the angle of elevation is 30o.

Solution:

We have,

h = 12

θ = 30o

Using the formula for shadow length, we have

s = h/tan θ

= 12/tan 30o

= 12/(1/√3)

= 20.78 m

Problem 4. Find the height of an object if the shadow length is 9 m and the angle of elevation is 55o.

Solution:

We have,

s = 9

θ = 55o

Using the formula for shadow length, we have

s = h/tan θ

=> 9 = h/tan 55o

=> h = 9 tan 55o

=> h = 9 (1.43)

=> h = 12.87 m

Problem 5. Find the height of an object if the shadow length is 14 m and the angle of elevation is 65o.

Solution:

We have,

s = 14

θ = 65o

Using the formula for shadow length, we have

s = h/tan θ

=> 14 = h/tan 65o

=> h = 14 tan 65o

=> h = 14 (2.144)

=> h = 30.01 m

Problem 6. Find the angle of elevation if the object has a height of 4 m and the shadow length is 9.42 m.

Solution:

We have,

h = 4

s = 9.42

Using the formula for shadow length, we have

s = h/tan θ

=> tan θ = h/s

=> tan θ = 4/9.42

=> tan θ = 0.42

=> θ = 23o

Problem 7. Find the angle of elevation if the object has a height of 16 m and a shadow length is 21.22 m.

Solution:

We have,

h = 16

s = 21.22

Using the formula for shadow length, we have

s = h/tan θ

=> tan θ = h/s

=> tan θ = 16/21.22

=> tan θ = 0.75

=> θ = 37o


Last Updated : 13 Mar, 2024
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