If the shadow of a tree increases by 3 meters when the angle of elevation of the sun’s rays decreases from 65°to 50°, find the height of the tree
make a sketch, label the position of the 50° angle as A
label the position of the 65° angle as B, where B is
clearly closer to the tree , The AB = 3 metres
label the top of tree as P and its bottom as Q
since angle PBQ = 65, angle ABP = 112°, which makes angle APB = 15°
by the sine law:
PB/sin50 = 3/sin15
PB = 3sin50/sin15 = ....
Now triangle PBQ is right-angled, so
so sin 65 = PQ/PB
PQ = height of tree = PBsin65
= (3sin50/sin15)(sin65) = ......
Draw a diagram and review your basic trig functions. If the height is h, then
h cot50° - h cot65° = 3
To find the height of the tree, we can use the concept of similar triangles.
Let's assume the original height of the tree is h meters, and the length of its shadow is x meters when the angle of elevation of the sun's rays is 65 degrees.
According to the given information, when the angle of elevation decreases to 50 degrees, the length of the shadow increases by 3 meters. Let's call this new shadow length y meters.
Now, we can set up a proportion using the similar triangles formed by the tree and its shadow.
Original height of the tree / Length of the shadow = Height of the tree when the angle of elevation decreases / Length of the new shadow
h / x = (h + 3) / y
Cross-multiplying, we get:
h * y = x * (h + 3)
Now, we can substitute the angles of elevation into trigonometric ratios to eliminate h and solve for y.
Using the trigonometric ratio for tangent:
tan(65°) = h / x
tan(50°) = (h + 3) / y
Now we can substitute these into the earlier equation:
x * tan(65°) = y * tan(50°)
Now, solve for y:
y = (x * tan(65°)) / tan(50°)
Substitute the known values:
y = (x * 1.428) / 1.192
Simplify:
y ≈ 1.2008x
Therefore, the height of the tree when the angle of elevation decreases to 50 degrees is approximately 1.2008 times the length of the new shadow.
In conclusion, the height of the tree is approximately 1.2008 times the increase in length of the shadow, which is 3 meters.
Therefore, the height of the tree is approximately 1.2008 * 3 = 3.6024 meters.
To find the height of the tree, we can use trigonometry and the concept of similar triangles.
Let's assume the initial height of the tree as 'h' meters, the length of the tree's shadow as 's' meters, and the increase in the shadow length as '3' meters.
We have two scenarios to work with:
1. In the first scenario, when the angle of elevation of the sun's rays is 65°, we have a right triangle formed by the tree, its shadow, and the sun's rays.
2. In the second scenario, when the angle of elevation decreases to 50°, we have another right triangle with an increased shadow length.
In both scenarios, the height of the tree forms the opposite side of the right triangle, and the length of the shadow forms the adjacent side.
Using trigonometry, we have:
1. In the first scenario:
tan(65°) = h / s
2. In the second scenario:
tan(50°) = (h + 3) / (s + 3)
We can now set up a system of equations to solve for 'h' and 's'.
From equation 1:
h = s * tan(65°)
Substituting this into equation 2:
tan(50°) = (s * tan(65°) + 3) / (s + 3)
Now, we can solve this equation to find the value of 's'.
Once we have the value of 's', we can substitute it back into equation 1 to find the height of the tree ('h').
Let me know if you would like me to solve this equation for you step by step.