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?

Explanation:

Step1: Identify the right - triangle relationship

We have a right - triangle where the height of the tree (opposite side with respect to the angle with the ground) is 10 m and the angle between the tree and the ground is $\theta = 60^{\circ}$, and we want to find the length of the shadow (adjacent side), let the length of the shadow be $x$.
We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
So, $\tan60^{\circ}=\frac{10}{x}$.

Step2: Solve for $x$

Since $\tan60^{\circ}=\sqrt{3}$, we have $\sqrt{3}=\frac{10}{x}$.
Cross - multiplying gives us $x = \frac{10}{\sqrt{3}}$.
Rationalizing the denominator by multiplying the numerator and denominator by $\sqrt{3}$, we get $x=\frac{10\sqrt{3}}{3}$.

Answer:

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