Question

a 12 foot ladder rests up against the side of the house. if an angle of 70° with the ground is recommended, how far from the house, in feet, should you put the bottom of the ladder?

Explanation:

Step1: Identify the trigonometric relation

We have a right triangle where the ladder is the hypotenuse (\(c = 12\) feet), the distance from the house is the adjacent side (\(x\)) to the angle \(\theta=70^{\circ}\), and we use the cosine function: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
So, \(\cos(70^{\circ})=\frac{x}{12}\)

Step2: Solve for \(x\)

Multiply both sides by 12: \(x = 12\times\cos(70^{\circ})\)
Calculate \(\cos(70^{\circ})\approx0.3420\)
Then \(x = 12\times0.3420\approx4.104\)

Answer:

Approximately \(4.10\) feet (or more precisely \(4.104\) feet)