when the angle of elevation is 30 degree the shadow of a vertical tower is 20m longer than when the elevation of the sun is 60 degree.find the height of the tower
To find the height of the tower, we can use trigonometry and set up a right triangle with the tower, the shadow, and the angle of elevation.
Let's assign variables to the known quantities:
- Let h be the height of the tower.
- Let x be the length of the shadow when the angle of elevation is 30 degrees.
- Let y be the length of the shadow when the angle of elevation is 60 degrees.
According to the problem, when the angle of elevation is 30 degrees, the shadow is 20m longer than when the angle of elevation is 60 degrees. So we have:
x = y + 20
Now, let's consider the right triangles formed in both cases.
When the angle of elevation is 30 degrees, we have the right triangle:
/|
/ |
/ |h
/ |
/ |
/θ |
--------
x
The adjacent side to the angle θ is x, and we know that tan(θ) = h/x.
When the angle of elevation is 60 degrees, we have the right triangle:
/|
/ |
/ |h
/ |
/ |
y / |
/θ |
--------
x
The opposite side to the angle θ is y, and tan(θ) = h/y.
Now, let's solve these equations to find the height of the tower (h):
From the equation above, we have x = y + 20.
Dividing both sides of the equation by y yields: x/y = 1 + 20/y.
Since tan(θ) = h/x, we can write: tan(30) = h/x and tan(60) = h/y.
Substituting these into the equation above, we get:
tan(30) = 1 + 20/y
Since tan(30) = 1/√3, we can rewrite the equation as:
1/√3 = 1 + 20/y
Now, we can solve for y:
1/y = 1/√3 - 1
y = 1/(1/√3 - 1)
Simplifying the expression on the right-hand side, we get:
y = √3 / (√3 - 1)
Now, we can substitute this value of y back into the equation x = y + 20 to find x:
x = (√3 / (√3 - 1)) + 20
Finally, we can substitute the values of x and y into either of the equations tan(30) = h/x or tan(60) = h/y to solve for the height of the tower (h).