Question

a ladder leans against the side of a house. the angle of elevation of the ladder is 61° when the bottom of the ladder is 12 ft from the side of the house. find the length of the ladder. round your answer to the nearest tenth.

Explanation:

Step1: Identify the trigonometric relationship

We know the adjacent - side to the angle of elevation and we want to find the hypotenuse. We use the cosine function, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 61^{\circ}$ and the adjacent - side $x = 12$ ft, and the hypotenuse is the length of the ladder $L$. So, $\cos(61^{\circ})=\frac{12}{L}$.

Step2: Solve for the length of the ladder

We can re - arrange the formula $\cos(61^{\circ})=\frac{12}{L}$ to $L=\frac{12}{\cos(61^{\circ})}$. Since $\cos(61^{\circ})\approx0.4848$, then $L=\frac{12}{0.4848}\approx24.75$.

Step3: Round the answer

Rounding $24.75$ to the nearest tenth gives $24.8$ ft.

Answer:

$24.8$ ft