Question

a ladder 10 m long rests against a vertical wall and makes an angle of 66°38 with the ground (assumed horizontal). the top of the ladder slips 2 m down the wall. what angle does the ladder now make with the ground?

Explanation:

Step1: Convert the initial angle to decimal degrees

We know that \(1^{\circ}=60'\), so \(38'=\frac{38}{60}\approx0.633^{\circ}\). Then \(66^{\circ}38'\approx66.633^{\circ}\)

Step2: Find the initial height of the top of the ladder

Using the sine function \(\sin\theta=\frac{opposite}{hypotenuse}\). Let the initial height be \(h_1\), with hypotenuse \(l = 10\)m and \(\theta=66.633^{\circ}\). So \(h_1 = 10\times\sin(66.633^{\circ})\approx10\times0.917 = 9.17\)m

Step3: Find the new height of the top of the ladder

The top slips \(2\)m down, so the new height \(h_2=h_1 - 2=9.17- 2 = 7.17\)m

Step4: Find the new angle \(\theta_2\)

Using the sine function again \(\sin\theta_2=\frac{h_2}{l}\), where \(h_2 = 7.17\)m and \(l = 10\)m. So \(\sin\theta_2=\frac{7.17}{10}=0.717\). Then \(\theta_2=\sin^{-1}(0.717)\approx45.89^{\circ}\)

Answer:

\(45.89^{\circ}\)