Final answer:
To determine the height of the tree, we use trigonometry with the given angle of 33 degrees and the distance of 100 feet from the tree. By applying the tangent function and multiplying by the distance, we find that the tree is approximately 64.94 feet tall.
Step-by-step explanation:
John wants to measure the height of a tree. He is standing 100 feet away from the tree and observes the angle of elevation to the top of the tree to be 33 degrees. To find the height of the tree, we can use trigonometry, specifically the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the opposite side is the height of the tree, and the adjacent side is the distance from John to the tree (100 feet).
The tangent of the angle of elevation (33 degrees) is equal to the height of the tree divided by the distance from John to the tree. Therefore, the formula to calculate the height of the tree is:
tan(33 degrees) = height of the tree / 100 feet
By rearranging the formula to solve for the height of the tree and using a calculator to find tan(33 degrees), we can calculate the height of the tree as follows:
height of the tree = tan(33 degrees) * 100 feet
When calculated, this gives us:
height of the tree = 0.6494 * 100 feet = 64.94 feet.
Therefore, John can estimate that the tree is approximately 64.94 feet tall. This is a practical application of trigonometry in real life, allowing us to measure heights that would be otherwise difficult to measure directly.