Question

a 14.5-m fire truck ladder is leaning against a wall. find the distance d the ladder goes up the wall (above the fire truck) if the ladder makes an angle of 27° 16 with the horizontal. d ≈ □ m (simplify your answer. type an integer or a decimal. round to the nearest hundredth.)

Explanation:

Step1: Convert angle to decimal degrees

First, convert \(16'\) to degrees. Since \(1^\circ = 60'\), then \(16'=\frac{16}{60}\approx0.2667^\circ\). So the angle \(\theta = 27^\circ+ 0.2667^\circ=27.2667^\circ\).

Step2: Use sine function

In a right triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the hypotenuse is the length of the ladder (\(14.5\) m) and the opposite side to the angle \(\theta\) is \(d\). So \(\sin(27.2667^\circ)=\frac{d}{14.5}\). Then \(d = 14.5\times\sin(27.2667^\circ)\).
Calculate \(\sin(27.2667^\circ)\approx\sin(27.27^\circ)\approx0.4583\) (using calculator). Then \(d = 14.5\times0.4583\approx6.64535\approx6.65\) (rounded to nearest hundredth).

Answer:

\(6.65\)