Question

a pine tree that is 10 m tall is damaged in a windstorm such that it leans sideways to make an angle of 60° with the ground. what is the exact expression for the length of the shadow of the tree when the sun is directly overhead?
$10\sqrt{3}$
5
$\frac{\sqrt{3}}{10}$
$\frac{1}{5}$

Explanation:

Step1: Set up a right - triangle model

The height of the tree is the opposite side of the right - triangle with respect to the angle it makes with the ground, and the shadow length is the adjacent side. Let the length of the shadow be $x$. We know the height of the tree $h = 10$ m and the angle $\theta=60^{\circ}$. We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan\theta=\tan(60^{\circ})=\frac{10}{x}$

Step2: Solve for $x$

Since $\tan(60^{\circ})=\sqrt{3}$, we have $\sqrt{3}=\frac{10}{x}$. Cross - multiply to get $x\sqrt{3}=10$, then $x = \frac{10}{\sqrt{3}}=\frac{10\sqrt{3}}{3}$. However, if we consider the right - triangle formed in another way (where the height of the tree is the hypotenuse and the shadow is the side adjacent to the $60^{\circ}$ angle), we use the cosine function. $\cos(60^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}$. Let the length of the shadow be $s$, and the height of the tree (hypotenuse) $H = 10$ m. Since $\cos(60^{\circ})=\frac{1}{2}$, and $\cos(60^{\circ})=\frac{s}{10}$, then $s = 5$.

Answer:

5