Question

a 14 - ft ladder leans against the side of a house. the angle of elevation of the ladder is 65°. how high is the top of the ladder from the ground? round your answer to the nearest tenth.

Explanation:

Step1: Identify the trigonometric relation

We have a right - triangle where the length of the ladder is the hypotenuse ($l = 14$ ft) and we want to find the height $h$ (opposite side) with respect to the given angle of elevation $\theta=65^{\circ}$. We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin\theta=\frac{h}{l}$

Step2: Substitute the values

Substitute $\theta = 65^{\circ}$ and $l = 14$ into the formula. $\sin(65^{\circ})=\frac{h}{14}$.
We know that $\sin(65^{\circ})\approx0.9063$. So, $h = 14\times\sin(65^{\circ})$.

Step3: Calculate the height

$h=14\times0.9063 = 12.6882\approx12.7$ ft.

Answer:

$12.7$ ft