Question

1. two trees in a park are 14.80 metres apart. an observer, whose eyeline is 1.80 m above the ground, is standing halfway between the trees. the angles of elevation of the tops of the trees from the observers eyeline are 17° and 23°. a) show this information on the diagram. b) determine the height of each tree to the nearest hundredth of a metre

Explanation:

Step1: Calculate the distance from the observer to each tree

The two trees are 14.80 metres apart and the observer is standing halfway between them. So the distance from the observer to each tree is $d=\frac{14.80}{2}=7.4$ metres.

Step2: Set up tangent - based equations for height calculation

Let $h_1$ and $h_2$ be the heights of the trees above the observer's eyeline. We know that $\tan\theta=\frac{opposite}{adjacent}$. For the first tree with angle of elevation $\theta_1 = 17^{\circ}$ and for the second tree with angle of elevation $\theta_2=23^{\circ}$, and the adjacent side (distance from observer to tree) $d = 7.4$ metres.
For the first tree: $h_1 = 7.4\times\tan(17^{\circ})$. Since $\tan(17^{\circ})\approx0.3057$, then $h_1=7.4\times0.3057 = 2.26218$ metres.
For the second tree: $h_2 = 7.4\times\tan(23^{\circ})$. Since $\tan(23^{\circ})\approx0.4245$, then $h_2=7.4\times0.4245 = 3.1413$ metres.

Step3: Calculate the actual heights of the trees

The observer's eyeline is 1.80 m above the ground.
The height of the first tree $H_1=h_1 + 1.80=2.26218+1.80=4.06218\approx4.06$ metres.
The height of the second tree $H_2=h_2 + 1.80=3.1413+1.80=4.9413\approx4.94$ metres.

Answer:

The height of the first tree is approximately 4.06 metres and the height of the second tree is approximately 4.94 metres.