Question

1. a 12 foot ladder leans against a wall, making an angle of 40° with the horizon. how high on the wall does the ladder rest?
2. a 12 foot ladder leans against a wall, making an angle of 40° with the horizon. how far is the base of the ladder from the wall?

set up trigonometric ratios and solve each problem. show calculations.

Explanation:

Step1: Analyze the right - triangle

The ladder forms a right - triangle with the wall and the ground. The length of the ladder is the hypotenuse \(c = 12\) feet.

Step2: Solve for the height on the wall (Question 1)

We use the sine ratio. \(\sin\theta=\frac{opposite}{hypotenuse}\), where \(\theta = 40^{\circ}\) and the opposite side \(y\) is the height on the wall.
\(\sin40^{\circ}=\frac{y}{12}\)
\(y = 12\times\sin40^{\circ}\)
Using a calculator, \(\sin40^{\circ}\approx0.6428\)
\(y\approx12\times0.6428 = 7.7136\) feet

Step3: Solve for the distance from the base of the ladder to the wall (Question 2)

We use the cosine ratio. \(\cos\theta=\frac{adjacent}{hypotenuse}\), where \(\theta = 40^{\circ}\) and the adjacent side \(x\) is the distance from the base of the ladder to the wall.
\(\cos40^{\circ}=\frac{x}{12}\)
\(x = 12\times\cos40^{\circ}\)
Using a calculator, \(\cos40^{\circ}\approx0.7660\)
\(x\approx12\times0.7660=9.192\) feet

Answer:

1. The ladder rests approximately \(7.71\) feet high on the wall.
2. The base of the ladder is approximately \(9.19\) feet from the wall.