Question

a 13.5-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 42° 45 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 \( 45' \) (minutes) to degrees. Since \( 1^\circ = 60' \), we have \( 45'=\frac{45}{60}=0.75^\circ \). So the angle \( \theta = 42^\circ + 0.75^\circ=42.75^\circ \).

Step2: Use sine function

We have a right triangle where the ladder is the hypotenuse (\( c = 13.5 \) m), and \( d \) is the opposite side to the angle \( \theta \). The sine function is defined as \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \), so \( \sin(\theta)=\frac{d}{13.5} \). Solving for \( d \), we get \( d = 13.5\times\sin(\theta) \).

Step3: Calculate \( \sin(42.75^\circ) \)

Using a calculator, \( \sin(42.75^\circ)\approx\sin(42.75)\approx0.6794 \).

Step4: Compute \( d \)

Substitute the values into the formula: \( d = 13.5\times0.6794 \approx 9.17 \) (rounded to the nearest hundredth).

Answer:

\( 9.17 \)