Question

a ladder 6 m long rests against a vertical wall and makes an angle of 68° 22 with the ground (assumed horizontal). the top of the ladder slips 1 m down the wall. what angle does the ladder now make with the ground?

Explanation:

Step1: Convert angle to decimal

First, convert \(68^{\circ}22'\) to decimal. Since \(1^{\circ} = 60'\), then \(22'=\frac{22}{60}\approx0.37^{\circ}\). So the initial angle \(\theta = 68.37^{\circ}\).

Step2: Find initial height

Using the sine function \(h = l\sin\theta\), where \(l = 6m\). So \(h=6\sin(68.37^{\circ})\). Using a calculator, \(\sin(68.37^{\circ})\approx0.93\), then \(h = 6\times0.93 = 5.58m\).

Step3: Find new height

The top slips \(1m\) down, so the new height \(h_{new}=5.58 - 1=4.58m\).

Step4: Find new angle

Using the sine function again \(\sin\alpha=\frac{h_{new}}{l}\), where \(l = 6m\) and \(h_{new}=4.58m\). So \(\sin\alpha=\frac{4.58}{6}\approx0.763\). Then \(\alpha=\sin^{- 1}(0.763)\approx49.72^{\circ}\).

Answer:

\(49.72^{\circ}\)