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

We consider a right - triangle where the height of the tree is the opposite side to the angle with the ground and the length of the shadow is the adjacent side. Let the length of the shadow be $x$. The height of the tree $h = 10$ m and the angle between the tree and the ground $\theta=60^{\circ}$.

Step2: Use the tangent function

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In our case, $\tan60^{\circ}=\frac{10}{x}$. Since $\tan60^{\circ}=\sqrt{3}$, we have the equation $\sqrt{3}=\frac{10}{x}$.

Step3: Solve for $x$

Cross - multiply to get $x\sqrt{3}=10$, then $x = \frac{10}{\sqrt{3}}$. Rationalize the denominator: $x=\frac{10\sqrt{3}}{3}$.

Answer:

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