The angle of elevation from a point on the ground to the top of a tree is 38.7 degrees. The angle of elevation from a point 27 ft farther back to the top of the tree is 22.9 degrees. Find the height of the tree to 2 decimal places.
draw a diagram
recall the definition of cotθ
then you can see that the height h is found using
h cot22.9° - h cot38.7° = 27
tan 30 degrees =x/25
To solve for the height of the tree, we can use trigonometry. Let's denote the height of the tree as 'h' and the distance between the first point and the tree as 'x'.
Using the given information, we have two right triangles:
In the first right triangle:
- The angle of elevation from the ground to the top of the tree is 38.7 degrees.
- The opposite side is 'h' (height of the tree).
- The adjacent side is 'x' (distance between the first point and the tree).
In the second right triangle:
- The angle of elevation from the point 27 ft farther back to the top of the tree is 22.9 degrees.
- The opposite side is 'h' (height of the tree).
- The adjacent side is 'x + 27' (distance between the second point and the tree).
Now, let's use trigonometry to write the equations:
In the first right triangle:
tan(38.7) = h / x
In the second right triangle:
tan(22.9) = h / (x + 27)
Now, we have two equations with two unknowns. We can solve this system of equations to find the values of 'h' and 'x'.
1. Rearrange the first equation to solve for h:
h = x * tan(38.7)
2. Substitute this value of h into the second equation:
tan(22.9) = (x * tan(38.7)) / (x + 27)
3. Cross multiply to get rid of the fractions:
(x + 27) * tan(22.9) = x * tan(38.7)
4. Expand and rearrange equation:
x * tan(22.9) + 27 * tan(22.9) = x * tan(38.7)
5. Rearrange to solve for x:
x * (tan(38.7) - tan(22.9)) = 27 * tan(22.9)
6. Divide both sides by (tan(38.7) - tan(22.9)):
x = (27 * tan(22.9)) / (tan(38.7) - tan(22.9))
After solving this equation, you will find the value of x.
Now, substitute this value of x back into the equation h = x * tan(38.7) to find the height of the tree 'h'.
Finally, round the value of h to 2 decimal places to get the height of the tree.