Question

if the angle of elevation of the sun is 69.5° when a building casts a shadow of 37.5 feet, what is the height of the building? round to at least the nearest tenth of a foot.

Explanation:

Step1: Set up the tangent - ratio

We know that in a right - triangle formed by the building, its shadow, and the line of sight to the sun, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the height of the building be $h$. The angle of elevation $\theta = 69.5^{\circ}$ and the length of the shadow (adjacent side) is $37.5$ feet. So, $\tan(69.5^{\circ})=\frac{h}{37.5}$.

Step2: Solve for $h$

Multiply both sides of the equation by $37.5$: $h = 37.5\times\tan(69.5^{\circ})$.
We know that $\tan(69.5^{\circ})\approx2.6694$. Then $h = 37.5\times2.6694$.
$h=37.5\times2.6694 = 100.1025$.

Step3: Round the answer

Rounding $100.1025$ to the nearest tenth, we get $h\approx100.1$.

Answer:

$100.1$