Question

question aiden leans a 30-foot ladder against a wall so that it forms an angle of 78° with the ground. whats the horizontal distance between the base of the ladder and the bottom of the wall? round your answer to the nearest hundredth of a foot if necessary. answer attempt 1 out of 2 feet submit answer

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the ladder is the hypotenuse (\(c = 30\) feet), the horizontal distance is the adjacent side (\(a\)) to the angle \(\theta=78^\circ\) with the ground. We use the cosine function: \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(78^\circ)=\frac{a}{30}\).

Step2: Solve for the adjacent side (horizontal distance)

Rearrange the formula to solve for \(a\): \(a = 30\times\cos(78^\circ)\). Calculate \(\cos(78^\circ)\approx0.2079\), then \(a = 30\times0.2079 = 6.237\). Round to the nearest hundredth: \(a\approx6.24\).

Answer:

\(6.24\)