Final answer:
Using trigonometry and the given angle of elevation of 59º at a distance of 45 feet from the tree, the Sitka spruce tree's estimated height is approximately 74.89 feet.
Step-by-step explanation:
To estimate the height of a Sitka spruce tree using the angle of elevation and distance from the tree, we use trigonometric ratios. Given the distance from the tree, which is 45 feet, and the angle of elevation, which is 59º, we can represent this situation as a right-angled triangle where the tree's height is the opposite side, and the distance from the tree is the adjacent side to the angle of elevation.
We use the tangent function, which relates the opposite side to the adjacent side in a right-angled triangle, defined as tangent of an angle = opposite / adjacent. Therefore, if we let ‘h’ be the height of the tree, the equation would be:
tan(59º) = h / 45 feet
To find the height ‘h’, solve for h:
h = 45 feet * tan(59º)
Now, calculating this using a calculator:
h ≈ 45 feet * 1.6643
h ≈ 74.8935 feet
Therefore, we estimate the height of the Sitka spruce tree to be approximately 74.89 feet.