Question

a 96 - ft tree casts a shadow that is 150 ft long. what is the angle of elevation of the sun? (round your answer to one decimal place.)

Explanation:

Step1: Set up the tangent - ratio

We can consider a right - triangle where the height of the tree is the opposite side and the length of the shadow is the adjacent side with respect to the angle of elevation of the sun. Let $\theta$ be the angle of elevation of the sun. The tangent of an angle in a right - triangle is given by $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the opposite side $a = 96$ ft and the adjacent side $b = 150$ ft. So, $\tan\theta=\frac{96}{150}$.

Step2: Simplify the tangent value

$\tan\theta=\frac{96}{150}=\frac{16}{25}=0.64$.

Step3: Find the angle

To find $\theta$, we take the inverse tangent (arctan) of 0.64. So, $\theta=\arctan(0.64)$. Using a calculator, $\theta=\arctan(0.64)\approx32.6^{\circ}$.

Answer:

$32.6$