Final answer:
To find the height of the tree, the tangent of the 40° angle of elevation is used to calculate the height above the woodcutter's eyes. By adding the woodcutter's eye level to the result of this calculation, we find that the tree is approximately 30 feet tall when the numbers are rounded to the nearest foot.
Step-by-step explanation:
To determine the height of the tall tree, we can use trigonometric functions. Specifically, we'll use the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. Here, the opposite side is the height of the tree above the woodcutter's eye level, and the adjacent side is the distance from the woodcutter to the tree, which is 30 feet.
The angle of elevation from the woodcutter's eyes to the top of the tree is 40°. The woodcutter's eyes are 5 feet above the ground. Using the tangent function:
- Write the tangent function for the angle of elevation:
tan(40°) = (tree height - 5 feet) / 30 feet - Solve for the tree height above the woodcutter's eyes (opposite side):
tree height - 5 feet = 30 feet × tan(40°)
- Add the woodcutter's eye level to the calculated height:
total tree height = (30 feet × tan(40°)) + 5 feet - Use a calculator to find tan(40°), which is approximately 0.8391.
- Multiply 0.8391 by 30 feet to get the height above the woodcutter's eyes.
- Add the woodcutter's eye level (5 feet) to find: The total height of the tree:
25.173 + 5 = 30.173 feet - Rounded to the nearest foot, the tree is approximately 30 feet tall.
Therefore, the height of the tree is approximately 30 feet when rounded to the nearest foot.