Question

a 15-foot ladder is leaning against a wall. if the ladder makes a 78° angle with the ground, how far is the base of the ladder from the wall?

Explanation:

Step1: Identify adjacent side, hypotenuse

We need the distance from the base of the ladder to the wall, which is the side adjacent to the 78° angle. The ladder length (15 ft) is the hypotenuse of the right triangle formed by the ladder, wall, and ground. We use the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Rearrange to solve for adjacent side

Isolate the adjacent side (let it be $x$):
$x = \text{hypotenuse} \times \cos(\theta)$
Substitute values: $x = 15 \times \cos(78^\circ)$

Step3: Calculate the value

Use a calculator to find $\cos(78^\circ) \approx 0.2079$:
$x \approx 15 \times 0.2079$
$x \approx 3.1185$

Answer:

Approximately 3.12 feet (or exactly $15\cos(78^\circ)$ feet)