Find the height of a tree given that it casts a shadow 1.5 m long when the elevation of the sun is 550 (see figure to the right).
Maybe you mean 55 degrees?
tan 55 = h/1.5
To find the height of the tree, we can use the concept of similar triangles. Here's how you can do it:
1. Draw a diagram: Start by drawing a diagram of the situation described. Label the height of the tree as 'h', the length of the shadow as 's', and the elevation of the sun as 'θ'.
2. Identify the similar triangles: In the diagram, you will notice that there are two right-angled triangles formed -- one with the tree and its shadow, and the other with the vertical height and the length of the shadow. These triangles are similar because they share the same angles.
3. Set up a proportion: Using the similar triangles, set up a proportion by equating the corresponding sides of the triangles. The corresponding sides are the height of the tree (h) and the length of the shadow (s), and the vertical height (h') and the length of the shadow (1.5m).
We can write the proportion as follows:
h / s = h' / 1.5m
4. Solve for the height of the tree: Substitute the values given in the problem into the proportion. We have the length of the shadow, s = 1.5m, and the angle of elevation of the sun, θ = 550.
h / 1.5m = h' / 1.5m
Since the length of the shadow in both triangles is the same, we can cancel them out:
h = h'
Therefore, the height of the tree is equal to the vertical height, which we need to find.
5. Find the vertical height: To find the vertical height (h'), we can use trigonometry. Since we know the length of the shadow (1.5m) and the angle of elevation of the sun (550), we can use the tangent function.
tan(550) = h' / 1.5m
Rearrange the equation to solve for h':
h' = 1.5m * tan(550)
Use a calculator to find the approximate value of tan(550). Multiply it by 1.5m to get the vertical height, h'.
6. Calculate the height of the tree: Since we determined earlier that the height of the tree is equal to the vertical height, h = h'.
Therefore, the height of the tree is approximately equal to the value of h' calculated in step 5.
By following these steps and using the given information, you can find the height of the tree.