Question

a pine tree that is 10 m tall is damaged in a wind - storm 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?

Explanation:

Step1: Set up a right - triangle model

The height of the tree is the vertical side of a right - triangle and the length of the shadow is the horizontal side. The tree forms an angle of $60^{\circ}$ with the ground. Let the length of the shadow be $x$. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 60^{\circ}$ and the opposite side to the angle with respect to the ground is the height of the tree ($h = 10$ m) and the adjacent side is the length of the shadow $x$.
$\tan60^{\circ}=\frac{10}{x}$

Step2: Solve for $x$

Since $\tan60^{\circ}=\sqrt{3}$, we can rewrite the equation as $\sqrt{3}=\frac{10}{x}$. Cross - multiplying gives us $x\sqrt{3}=10$. Then, solving for $x$ we get $x = \frac{10}{\sqrt{3}}$. Rationalizing the denominator (multiplying numerator and denominator by $\sqrt{3}$), we have $x=\frac{10\sqrt{3}}{3}$.

Answer:

$\frac{10\sqrt{3}}{3}$ m