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The shadow of the tower standing on a level ground is x m long when the Sun's altitude is 30 , while it is y m long when the altitude is 60 . If the height of the tower is 4532 m, then x−y is

Answer
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Hint: Here, we will make a diagram according to the given information, so that we can understand the question more properly. Then by using Pythagoras theorem, we can get the value of tan = Opposite/ Adjacent. Using this tan value, we can find the values of x and y. Finally, we will substitute the values to get the final output of x – y.

Complete step-by-step answer:
According to the given information in the question, we can make the figure as below:
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Let the given tower be AD and the height of the tower is given.
 AD=4532 m
Given that,
When the angle of elevation is 30 , the length of the shadow is x metres.
 DCA=30 and AC = x metres.
Also, given that,
When the angle of elevation is 60 , the length of the shadow is y metres.
 DBA=60 and AB = y metres.
We know that, tan C = side opposite to angle C / side adjacent to angle C
Now, we will find the values of x and y to get the final output required.

First we will find the value of x:
In right angle triangle ΔDAC ,
 tanC=ADAC
Here, the angle of elevation is 30 and so substituting the values we will get,
 tan30=4532x
 tan30=453x×2
Since we know that tan30=13 .
Substituting the value of tan we will get,
 13=453x×2
On evaluating the above equation, we will get,
 x=45313×2
 x=453÷23
 x=453×32
 x=45(3)2 (3×3=3)
 x=1352

Second, we will find the value of y:
In right angle triangle ΔDAB
 tanB=ADAB
Here, the angle of elevation is 60 and so substituting the values we will get,
 tan60=453y×2
We know that tan60=4532y
Since we know that tan60=3 .
Substituting the value of tan in the above equation, we will get,
 3=453y×2
On evaluating the above equation, we will get,
 y=4533×2
 y=452
Lastly we need to find the value of x – y:
 xy
Substituting the values of x and y in the above expression, we will get,
 =1352452
Taking LCM as 2 , we will get,
 =135452
Simplify this above expression, and it is reduce in the following form
 =902
 =45 metres
Hence, if the height of the tower is 4532 m, then the value of xy=45 m.
So, the correct answer is “ xy=45 m.”.

Note: Pythagoras’ Theorem describes the mathematical relationship between three sides of a right-angled triangle. Trigonometry is a field of study in mathematics which observes the relationships of the sides and angles of triangles. The symbol θ is used to describe an unknown angle. The trigonometric functions sine, cosine and tangent, or as they are more commonly known as sin, cos and tan, can be used to find missing sides or angles of triangles. These functions are defined as the ratios of the different sides of a triangle.
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