Question

a ladder is leaning against a wall. the angle of elevation from the ground to the top of the ladder is 56°. the top of the ladder is 8 feet above the ground. what is the length of the ladder? 14.4 feet 9.6 feet 14.3 feet 14.1 feet 5.4 feet 13.1 feet

Explanation:

Step1: Identify the trigonometric relationship

We know the height (opposite side to the given angle) and we need to find the length of the ladder (hypotenuse). We use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 56^{\circ}$ and the opposite side $y = 8$ feet. Let the length of the ladder (hypotenuse) be $x$. So, $\sin(56^{\circ})=\frac{8}{x}$.

Step2: Solve for \(x\)

We can rewrite the equation as $x=\frac{8}{\sin(56^{\circ})}$. Using a calculator, $\sin(56^{\circ})\approx0.829$. Then $x=\frac{8}{0.829}\approx9.6$.

Answer:

$9.6$ feet