A tree is growing at an angle out of the ground, 15°17' from vertical. Standing 10 ft away from the base, the angle to the top of the tree is 25°17'. What is the height of the tree?
To determine the height of the tree, we can use trigonometry. Let's break down the problem step by step and explain how to get the answer:
Step 1: Draw a diagram
Start by drawing a diagram to understand the given information. Draw a right-angled triangle where the tree is the vertical side, the distance from the base to the top of the tree is the hypotenuse, and the angle between the vertical side and the hypotenuse is 25°17'. Label the vertical side as "h" (height of the tree) and the hypotenuse as "d" (distance from the base to the top of the tree).
Step 2: Find the angle between the tree and the vertical
Since the tree is growing at an angle out of the ground, 15°17' from vertical, subtract this angle from 90° to find the angle between the tree and the vertical. In this case, the angle is 90° - 15°17', which equals 74°43'.
Step 3: Use trigonometry to find the height of the tree
We have a right-angled triangle with the angle (74°43') between the vertical side (h) and the hypotenuse (d). We can use the trigonometric function tangent (tan) to find the height of the tree.
tan(74°43') = h / 10 ft
Rearrange the equation to solve for h:
h = 10 ft * tan(74°43')
Now we can calculate the value using a scientific calculator or by using a trigonometric table.
Step 4: Evaluate the equation to find the height of the tree
Using a calculator or trigonometric table, find the tangent of 74°43' and multiply it by 10 ft:
h = 10 ft * tan(74°43')
After evaluating the equation, you will get the height of the tree.
Depends - is the tree leaning toward you or away from you?
or, for an even more interesting scenario ...
the tree could be leaning sideways.