Question

olivia leans a 28-foot ladder against a wall so that it forms an angle of 60° 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 trigonometric ratio

The height \(x\) is opposite the \(60^\circ\) angle; ladder (28ft) is hypotenuse. Use \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\).
\(\sin(60^\circ) = \frac{x}{28}\)

Step2: Solve for \(x\)

Rearrange: \(x = 28 \times \sin(60^\circ)\). \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\).
\(x = 28 \times \frac{\sqrt{3}}{2} = 14\sqrt{3}\)

Step3: Calculate numerical value

\(\sqrt{3} \approx 1.73205\), so \(14 \times 1.73205 \approx 24.2487\)

Step4: Round to nearest hundredth

\(24.2487 \approx 24.25\)

Answer:

24.25