The shadow of a tower when the angle of elevation of the sun is 45 degree is found to be 5m longer when it is 60 degree. Find the height of the tower.
sin45=h/(d)
sin60=h/(d+5)
h/d=.707
h/(d+5)=.866
d=h*1.414
so
h=(h*1.414+5)*.866
solve for h
the wording of the problem is awkward, but a steeper angle means a shorter shadow
h/(s + 5) = tan(45º) ... h = s + 5
... s = h - 5
h / s = tan(60º)
h = h tan(60º) - 5 tan(60º)
5 tan(60º) = h [tan(60º) - 1)
The distance should be 5m LONGER at 45o.
Tan 45 = h/(d+5).
h = (d+5)*Tan45 = d+5.
Tan60 = h/d.
h = d*Tan60 = 1.73d
d+5 = 1.73d.
d = 6.83 m.
h = d + 5 = 6.83+5 = 11.83 m.
I agree.
To find the height of the tower, we need to use trigonometry and set up some equations based on the given information.
Let's assume the height of the tower is "h" meters.
When the angle of elevation of the sun is 45 degrees, the shadow of the tower is found to be 5 meters longer than when it is 60 degrees.
Let's denote the length of the shadow at 45 degrees as "x" meters.
According to trigonometry, we can set up the following equation:
tan(45 degrees) = h / x (1)
Similarly, for the angle of elevation of 60 degrees, let's denote the length of the shadow as "x + 5" meters.
So we can set up another equation:
tan(60 degrees) = h / (x + 5) (2)
Using the values of tangent for 45 degrees and 60 degrees:
√2 = h / x (3)
√3 = h / (x + 5) (4)
Now we can solve equations (3) and (4) simultaneously to find the height of the tower "h".
First, let's isolate "h" in equation (3):
h = x * √2 (5)
Now substitute the value of "h" from equation (5) into equation (4):
√3 = (√2 * x) / (x + 5)
Cross-multiply and simplify:
√3 * (x + 5) = √2 * x
√3 * x + √3 * 5 = √2 * x
Simplify further:
√3 * x - √2 * x = - √3 * 5
To solve this equation, we can isolate "x" by moving one term to the other side:
√3 * x - √2 * x = - √3 * 5
(x * √3 - x * √2) = -√3 * 5
Factor out "x" on the left side:
x * (√3 - √2) = -√3 * 5
Divide both sides by (√3 - √2):
x = (-√3 * 5) / (√3 - √2)
Now we can use this value of "x" in equation (5) to find the height of the tower "h":
h = x * √2
Substitute the value of "x" we just calculated and simplify:
h = [(-√3 * 5) / (√3 - √2)] * √2
Therefore, the height of the tower is (-5√6) / (√3 - √2) meters.