Question

a ladder 7 m long rests against a vertical wall and makes an angle of 56°35 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 the angle \(56^{\circ}35'\) to decimal degrees

Since \(1^{\circ} = 60'\), then \(35'=\frac{35}{60}\approx0.583^{\circ}\). So \(56^{\circ}35'\approx56.583^{\circ}\)

Step2: Find the initial height of the top - of - the - ladder on the wall

Using the sine function \(y = l\sin\theta\), where \(l = 7m\) and \(\theta=56.583^{\circ}\). So \(y = 7\sin(56.583^{\circ})\approx7\times0.835 = 5.845m\)

Step3: Find the new height of the top - of - the - ladder on the wall

The top of the ladder slips \(1m\) down, so the new height \(y_{new}=5.845 - 1=4.845m\)

Step4: Find the new angle \(\alpha\)

Using the sine function again \(\sin\alpha=\frac{y_{new}}{l}\), with \(y_{new} = 4.845m\) and \(l = 7m\). So \(\sin\alpha=\frac{4.845}{7}\approx0.692\)

Then \(\alpha=\sin^{- 1}(0.692)\approx43.82^{\circ}\)

Answer:

\(43.82^{\circ}\)