Question

a 25 - foot long ladder is propped against a wall at an angle of 18° with the wall. how high up the wall does the ladder reach? round the answer to the nearest tenth of a foot. 13.8 ft 23.8 ft 26.3 ft 80.9 ft

Explanation:

Step1: Identify the trigonometric relationship

We know the length of the ladder (hypotenuse $c = 25$ ft) and the angle between the ladder and the wall $\theta=18^{\circ}$. We want to find the height $h$ up the wall the ladder reaches. We use the cosine - function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, the height up the wall is the adjacent side to the angle $\theta$ with respect to the right - triangle formed by the ladder, the wall, and the ground. So, $\cos\theta=\frac{h}{c}$.

Step2: Solve for $h$

We can rewrite the formula as $h = c\times\cos\theta$. Substitute $c = 25$ ft and $\theta = 18^{\circ}$ into the formula. Since $\cos(18^{\circ})\approx0.9511$, then $h=25\times\cos(18^{\circ})$.
$h = 25\times0.9511=23.7775$ ft.

Step3: Round the answer

Rounding $23.7775$ ft to the nearest tenth of a foot gives $h\approx23.8$ ft.

Answer:

$23.8$ ft