When the angle of elevation of the sun is 30° the shadow of a vertical tower is 20m longer than when the elevation of the sun is 60°.find the height of the tower
After making your sketch, .....
Let the shadow's length at 60° be x m, let the tower height be h
Given: the length of the shadow at 30° = x+20 m
so ... tan60 = h/x
h = xtan60
and tan 30 = h/(x+20)
h = tan30(x+20)
then xtan60 = tan30(x+20)
xtan60 = xtan30 + 20tan30
x(tan60 - tan30) = 20tan30
x = 20tan30/((tan60 - tan30) = ....
then you find h in h = xtan60
OR
Nice to have those 30° and 60° angles, since you have the 30-60-90 triangle
with corresponding sides in the ratio of 1:√3:2
By the 30-60-90 ratios:
√3 x / x = (x+2)/(√3x)
√3 = (x+20)/(√3x)
3x = x+20
x = 10
then h = √3x = √3(10) = appr 17.321.... m
You will get the same answer from my first solution.
Reiny's first solution can be made at least to look less complicated if you are comfortable using the cotangent function.
h cot30° - h cot60° = 20
h = 20/(cot30° - cot60°) = 20/(√3 - 1/√3)
I need the real solution to the question
i need the real solution t
o this answer pls
To find the height of the tower, we can set up a trigonometric equation using the given information.
Let's assume the height of the tower is represented by 'h' meters.
When the angle of elevation of the sun is 30°, the shadow of the tower is 20 meters longer. This means that the length of the shadow is 'h + 20' meters.
When the angle of elevation of the sun is 60°, the length of the shadow is 'h' meters.
We can use the tangent function to relate the angle of elevation and the height of the tower:
tan(30°) = (h + 20) / length of shadow when angle = 30° ---> Equation 1
tan(60°) = h / length of shadow when angle = 60° ---> Equation 2
To solve for 'h', we can rearrange Equation 2 to get:
h = length of shadow when angle = 60° * tan(60°)
Now, we can substitute this value of 'h' into Equation 1 to solve for the length of the shadow when the angle of elevation is 30°:
tan(30°) = (length of shadow when angle = 60° * tan(60°) + 20) / length of shadow when angle = 30°
Simplifying this equation will give us the length of the shadow when the angle of elevation is 30°.
Finally, we can subtract the length of the shadow from the height of the tower to find the value of 'h'.