Calculate the height of a tree whose shadow is 20m shorter when the angle of elevation of the sun is 60° than when the angle of elevation is 30°
Draw a diagram and review the basic trig functions. You can see that if the height is h, then
h cot30° - h cot60° = 20
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Calculate the height of a tree whose shadow is 20 metres shorter when the angle of elevation of the sun is 60° than when the angle of elevation is 30°
To solve this problem, you can use trigonometry. Let's consider the angle of elevation of the sun at 30° and 60°. Let's assume the height of the tree is represented by 'h.'
When the angle of elevation is 30°, we can create a right-angled triangle with the height of the tree as the opposite side, the shadow length as the adjacent side, and the angle of elevation as the angle opposite the height of the tree. This gives us the equation:
tan(30°) = h / shadow length
Simplifying this equation, we have:
1 / √3 = h / shadow length
When the angle of elevation is 60°, we have another right-angled triangle with the height of the tree as the opposite side, the shadow length reduced by 20m as the adjacent side, and the angle of elevation as the angle opposite the height of the tree. This gives us the equation:
tan(60°) = h / (shadow length - 20)
Simplifying this equation, we have:
√3 = h / (shadow length - 20)
Now, we can set up a system of equations by combining both equations:
1 / √3 = h / shadow length
√3 = h / (shadow length - 20)
To solve this system, we can isolate 'h' in both equations:
h = shadow length / √3
h = √3 * (shadow length - 20)
Since both expressions equal 'h', we can set them equal to each other:
shadow length / √3 = √3 * (shadow length - 20)
Simplifying this equation, we have:
shadow length = √3 * (shadow length - 20) * √3
shadow length = 3 * (shadow length - 20)
shadow length = 3 * shadow length - 60
2 * shadow length = 60
shadow length = 30
Now that we know the shadow length is 30m, we can substitute this value into either of the original equations to find the height of the tree. Using the first equation:
h = 30 / √3
h ≈ 17.32 m
Therefore, the height of the tree is approximately 17.32 meters.