Question

Dawn and Gemma are standing on opposite sides of a tree. Dawn is at point D which is 15 meters from the base of the tree. The angle of elevation where she stands is 30 degrees. Gemma is standing at point G. She is k meters from the base of the tree. The angle of elevation where she stands to the top of the tree is at 45 degrees. Find the exact height of the tree in simplest radical form. Calculate Gemma’s distance from the base of the tree give answer in simplest radical form.

Answer 1

Using trigonometric functions based on the angles of elevation provided, the height of the tree is found to be 5√3 meters. Additionally, Gemma's distance from the base of the tree is calculated to be the same, 5√3 meters, due to her 45 degrees angle of elevation.

Step-by-step explanation:

We have two scenarios in this trigonometry problem. Dawn is 15 meters away from the tree with a 30 degrees angle of elevation, while Gemma has a 45 degrees angle of elevation, standing at a distance of k meters from the tree.

To find the height of the tree using the information given for Dawn, we can use the basic trigonometric function tan(θ) = opposite/adjacent. Here, tan(30 degrees) = height/15. Since tan(30 degrees) = 1/√3, we can solve for the height (h) of the tree: 1/√3 = h/15, which yields h = 15/√3 = 5√3 meters.

For Gemma, since the angle of elevation is 45 degrees, and tan(45 degrees) = 1, we have the tree's height being equal to her distance from the base of the tree. Therefore, k is also 5√3 meters.