A tree casts a 35 meter long shadow at a time when the sun is 41° above the horizon. How tall is the tree?
Would the answer be 53 meters?
Try drawing a figure. Height is what we are finding, base of triangle is 35 meters. The angle theta is 41 degrees. Use the definition of tangent. tan(41 degrees) = height/35
So, no the answer is not 53 meters. Rather, it is approximately 30.425 m
To find the height of the tree, we can use the concept of similar triangles.
Since the sun is 41° above the horizon, we can consider the triangle formed by the tree, its shadow, and a line from the top of the tree to the sun. This forms two similar triangles - one larger triangle involving the tree, its shadow, and the line from the top of the tree to the sun, and another smaller triangle formed by the tree, its shadow, and the ground.
Let's call the height of the tree "h" and the length of the shadow "s". The angle between the ground and the line from the top of the tree to the sun is 90° - 41° = 49°.
In the larger triangle, we have:
(h + s)/h = tan(49°)
In the smaller triangle, we have:
s/h = tan(41°)
Since we want to find the height of the tree, h, we can rearrange the equations:
From the larger triangle:
(h + s)/h = tan(49°)
h + s = h * tan(49°)
From the smaller triangle:
s/h = tan(41°)
s = h * tan(41°)
Now we can substitute the value of s from the smaller triangle into the equation from the larger triangle:
h + h * tan(41°) = h * tan(49°)
h * (1 + tan(41°)) = h * tan(49°)
Dividing both sides by h, we get:
1 + tan(41°) = tan(49°)
Now we can solve for h:
h = (tan(49°))/(1 + tan(41°))
Plugging in the values into a calculator, we find that h ≈ 53.12 meters.
So, the height of the tree is approximately 53.12 meters, not 53 meters.