Question

parker leans a 16-foot ladder against a wall so that it forms an angle of 64° with the ground. how high up the wall does the ladder reach? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle where the ladder is the hypotenuse (16 feet), the angle with the ground is \(64^\circ\), and we need to find the height (opposite side to the angle). So we use the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)

Step2: Substitute the values

Let \(x\) be the height. Then \(\sin(64^\circ)=\frac{x}{16}\)

Step3: Solve for \(x\)

Multiply both sides by 16: \(x = 16\times\sin(64^\circ)\)
Calculate \(\sin(64^\circ)\approx0.8988\), so \(x\approx16\times0.8988 = 14.3808\)

Step4: Round to nearest hundredth

\(x\approx14.38\)

Answer:

The ladder reaches approximately \(\boxed{14.38}\) feet up the wall.