Question

riley leans a 16-foot ladder against a wall so that it forms an angle of 74° 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.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the ladder is the hypotenuse (16 feet), the horizontal distance is the adjacent side to the angle of \(74^\circ\), and we can use the cosine function. The cosine of an angle in a right triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Set up the equation

Let \(x\) be the horizontal distance (adjacent side). Then \(\cos(74^\circ)=\frac{x}{16}\).

Step3: Solve for \(x\)

Multiply both sides of the equation by 16 to isolate \(x\): \(x = 16\times\cos(74^\circ)\).
We know that \(\cos(74^\circ)\approx0.2756\) (using a calculator). So \(x = 16\times0.2756\).

Step4: Calculate the value

\(16\times0.2756 = 4.4096\approx4.41\) (rounded to the nearest hundredth).

Answer:

The horizontal distance is approximately \(4.41\) feet.