Question

as shown in the diagram below, a ladder 5 feet long leans against a wall and makes an angle of 65° with the ground. find, to the nearest tenth of a foot, the distance from the wall to the base of the ladder

Explanation:

Step1: Identify the trigonometric ratio

We know the hypotenuse (\(c = 5\) feet) and we need to find the adjacent side (\(x\)) to the angle \(\theta=65^{\circ}\). The cosine ratio is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(65^{\circ})=\frac{x}{5}\).

Step2: Solve for \(x\)

Multiply both sides of the equation \(\cos(65^{\circ})=\frac{x}{5}\) by \(5\). We get \(x = 5\times\cos(65^{\circ})\).
Using a calculator, \(\cos(65^{\circ})\approx0.4226\). Then \(x = 5\times0.4226=2.113\).

Answer:

\(2.1\) feet