Question

a 12 - foot ladder is placed 4 feet away from a house as shown. what is the value of x, rounded to the nearest degree? according to the manufacturer, if the angle the ladder forms with the ground, x, is between 70° and 75° then it meets safety standards. is this ladder being used safely?

Explanation:

Step1: Use cosine function

We know that in a right - triangle, $\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, the hypotenuse of the right - triangle formed by the ladder, the ground and the house is the length of the ladder $l = 12$ ft and the adjacent side to the angle $x$ is the distance from the house to the base of the ladder $a = 4$ ft. So, $\cos x=\frac{4}{12}=\frac{1}{3}$.

Step2: Find the angle

We need to find $x$, and since $\cos x=\frac{1}{3}$, then $x=\cos^{- 1}(\frac{1}{3})$. Using a calculator, $x\approx70.53^{\circ}$.

Answer:

$x\approx71^{\circ}$

To check safety: Since the manufacturer states that the angle $x$ should be between $70^{\circ}$ and $75^{\circ}$, and $70^{\circ}<71^{\circ}<75^{\circ}$, the ladder is being used safely.