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.
How about starting with a diagram.
Label the tower as PQ, with Q on the ground.
Label the point with angle 45° as A, and the point with the 60° angle as B.
You now have two right-angled triangles, PAQ and PBQ
let BQ be x
then tan60 = PQ/x ---> PQ = xtan60 = √3 x
and tan 45 = PQ/(x+5) ---> PQ = (x+5)tan45 = (x+5)(1)
so √3 x =x+5
√3 x - x = 5
x(√3 - 1) = 5
x = 5/(√3-1)
then PQ = √3(5/(√3-1)) = appr 11.83 m
or
look at triangle PAB, angle PBA = 120° making angle APB = 15°
by the sine law:
PB/sin45 = 5/sin15
PB = 5sin45/sin15
in triangle PBQ,
sin60 = PQ/PB
PQ = PBsin60 = (5sin45/sin15)(√3/2) = appr 11.83
In my experience, most students find the second method easier to follow
Ans is correct can you show me fig.
We can't put diagrams on here, I think you should be able to follow my directions. I described the diagram.
To find the height of the tower, we can use trigonometry and set up two equations based on the given angles of elevation.
Let's assume the height of the tower is represented by "h" meters and the length of the shadow in the first scenario (angle of elevation = 45 degrees) is represented by "x" meters.
In the first scenario (angle of elevation = 45 degrees), we can form a right triangle with the height of the tower, the length of the shadow, and the angle of elevation. Based on trigonometry, the tangent of the angle of elevation is equal to the height of the tower divided by the length of the shadow:
tan(45 degrees) = h / x
In the second scenario (angle of elevation = 60 degrees), the length of the shadow is 5 meters longer. Therefore, the length of the shadow in this scenario can be represented as "x + 5" meters.
Again, we can form a right triangle with the height of the tower, the length of the shadow, and the angle of elevation. Using trigonometry, the tangent of the angle of elevation is equal to the height of the tower divided by the length of the shadow:
tan(60 degrees) = h / (x + 5)
Now, we have two equations with two unknowns (h and x):
Equation 1: tan(45 degrees) = h / x
Equation 2: tan(60 degrees) = h / (x + 5)
To find the height of the tower, we can solve these equations simultaneously.
First, let's substitute the values of the tangents of the angles:
Equation 1: 1 = h / x
Equation 2: √3 = h / (x + 5)
Now, we can solve these equations. By rearranging Equation 1, we have:
h = x
Substituting this value in Equation 2, we get:
√3 = x / (x + 5)
Cross-multiplying, we have:
√3 (x + 5) = x
Expanding the left side, we get:
√3x + 5√3 = x
Bringing all the terms on one side, we have:
x - √3x = -5√3
Factoring out x, we get:
x (1 - √3) = -5√3
Dividing both sides by (1 - √3), we get:
x = -5√3 / (1 - √3)
Simplifying further, we multiply the numerator and denominator by (1 + √3):
x = (-5√3)(1 + √3) / (1 - √3)(1 + √3)
Expanding, we get:
x = (-5√3 - 15) / (-2)
Simplifying, we have:
x = (15 + 5√3) / 2
Now, we substitute this value of x back into Equation 1 to find the height of the tower:
1 = h / [(15 + 5√3) / 2]
Simplifying, we get:
h = [(15 + 5√3) / 2]
Therefore, the height of the tower is (15 + 5√3) / 2 meters.