Question

1. a 30 foot ladder forms an angle of ( 50^{circ} ) with the ground when place against a brick wall. how far up the wall will the ladder reach? how far away from the base of the wall is the ladder?

Explanation:

Step1: Find how far up the wall the ladder reaches (height \(h\))

We use the sine function. In a right - triangle (where the ladder is the hypotenuse \(c = 30\) feet, the height on the wall is the opposite side to the angle \(\theta=50^{\circ}\)). The formula for sine is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).
So, \(\sin(50^{\circ})=\frac{h}{30}\).
Then \(h = 30\times\sin(50^{\circ})\).
Using a calculator, \(\sin(50^{\circ})\approx0.7660\), so \(h=30\times0.7660 = 22.98\) feet.

Step2: Find the distance \(d\) from the base of the wall

We use the cosine function. In a right - triangle, the distance from the base of the wall is the adjacent side to the angle \(\theta = 50^{\circ}\). The formula for cosine is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).
So, \(\cos(50^{\circ})=\frac{d}{30}\).
Then \(d = 30\times\cos(50^{\circ})\).
Using a calculator, \(\cos(50^{\circ})\approx0.6428\), so \(d=30\times0.6428=19.284\) feet.

Answer:

The ladder reaches approximately \(23.0\) feet up the wall and is approximately \(19.3\) feet away from the base of the wall.