Question

a pine tree that is 10 m tall is damaged in a windstorm such that it leans sideways to make an angle of 60^0 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 is the hypotenuse ($h = 10$ m) and the length of the shadow is the side adjacent to the given angle $\theta=60^{\circ}$. We use the cosine function $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
Let the length of the shadow be $x$. Then $\cos\theta=\frac{x}{h}$.

Step2: Substitute the values

Given $\theta = 60^{\circ}$ and $h = 10$ m, and $\cos60^{\circ}=\frac{1}{2}$. Substituting into the formula $\cos\theta=\frac{x}{h}$, we get $\frac{1}{2}=\frac{x}{10}$.

Step3: Solve for $x$

Cross - multiply to find $x$. We have $x = 10\times\cos60^{\circ}$. Since $\cos60^{\circ}=\frac{1}{2}$, then $x = 5$ m.

Answer:

$5$ m