Question

tantila leans a 30 - foot ladder against a wall so that it forms an angle of ( 65^{circ} ) 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 relationship

We have a right - triangle where the ladder is the hypotenuse (\(c = 30\) feet), the height on the wall is the opposite side (\(a\)) to the given angle \(\theta=65^{\circ}\). Using the sine function \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin(65^{\circ})=\frac{a}{30}\).

Step2: Solve for \(a\)

Multiply both sides of the equation \(\sin(65^{\circ})=\frac{a}{30}\) by \(30\). We know that \(\sin(65^{\circ})\approx0.9063\). Then \(a = 30\times\sin(65^{\circ})\).

$$a=30\times0.9063 = 27.189$$

Answer:

\(27.19\) feet