Question

a tree casts a shadow of 268 feet. if the angle of elevation is 47°, what is the distance from the tip of the shadow to the top of the tree? (round to 1 decimal place)

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the length of the adjacent side to the angle of elevation is the length of the shadow ($a = 268$ feet) and we want to find the hypotenuse ($c$). We use the cosine function: $\cos\theta=\frac{a}{c}$, where $\theta = 47^{\circ}$ and $a$ is the adjacent side and $c$ is the hypotenuse.

Step2: Rearrange the formula to solve for $c$

From $\cos\theta=\frac{a}{c}$, we can rewrite it as $c=\frac{a}{\cos\theta}$.

Step3: Substitute the values

Substitute $a = 268$ and $\theta = 47^{\circ}$ into the formula. $\cos(47^{\circ})\approx0.682$. So $c=\frac{268}{0.682}$.

Step4: Calculate the value

$c=\frac{268}{0.682}\approx393.0$

Answer:

$393.0$