Question

1. a ladder that is 20ft long is leaning against the side of a building. if the angle formed between the ladder and the ground is 75°, how far is the bottom of the ladder from the base of the building?

Explanation:

Step1: Identify the trigonometric relationship

We know the length of the hypotenuse (ladder length \(c = 20\) ft) and the angle \(\theta=75^{\circ}\), and we need to find the adjacent side \(x\) (distance of the bottom of the ladder from the base of the building). Using the cosine function: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(x = c\times\cos\theta\).

Step2: Substitute the values

Substitute \(c = 20\) and \(\theta = 75^{\circ}\) into the formula. We know that \(\cos(75^{\circ})=\cos(45^{\circ}+ 30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.2588\). Then \(x = 20\times\cos(75^{\circ})\).

$$x=20\times0.2588 = 5.176$$

Answer:

The bottom of the ladder is approximately \(5.18\) ft from the base of the building.