Final answer:
To determine the height of the tree, two right-angled triangles are formed by stepping back 10 feet between measurements of the angle of elevation. These triangles are solved simultaneously using the tangent ratios and angles given, resulting in the height of the tree.
Step-by-step explanation:
Determining the Height of a Tree Using Angles of Elevation
The question asks for the calculation of the height of a tree using the angles of elevation from two different points on the ground. This is a trigonometry problem that involves the concepts of right-angled triangles and tangent ratios. When you step back 10 feet from the initial measurement point, you create two right-angled triangles which share a common vertical side - the height of the tree. We will label this height as 'h'.
The first triangle is formed by the height of the tree, the distance from the original point to the base of the tree, and the line of sight from the point to the top of the tree at a 35.78° angle of elevation. Let's call this distance 'd'. According to the tangent function, we have:
tan(35.78°) = h / d
The second triangle has an angle of elevation of 27.7° and the distance from this point to the base of the tree is 'd + 10' feet. The tangent function for this triangle is:
tan(27.7°) = h / (d + 10)
We have two equations with two variables (h and d). We can solve these equations simultaneously to find the values of 'h' and 'd'. Calculating the value of 'h' will give us the height of the tree.
Note that to solve this problem, we will need to use a calculator to find the tangent values and to perform the calculations for h and d.