Question

solve each problem. see examples 1 - 4. 45. height of a ladder on a wall a 13.5 - m fire truck ladder is leaning against a wall. find the distance d the ladder goes up the wall (above the top of the fire truck) if the ladder makes an angle of 43°50 with the horizontal.

Explanation:

Step1: Convert angle to decimal degrees

First, convert $43^{\circ}50'$ to decimal - degrees. Since $1^{\circ}=60'$, then $50'=\frac{50}{60}\approx0.833^{\circ}$, so $43^{\circ}50'\approx43.833^{\circ}$.

Step2: Use sine function

We know that in a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, the hypotenuse is the length of the ladder $l = 13.5$ m and the angle $\theta\approx43.833^{\circ}$, and the opposite side to the angle $\theta$ is the height $d$ we want to find. So, $\sin\theta=\sin(43.833^{\circ})=\frac{d}{13.5}$.

Step3: Solve for $d$

Multiply both sides of the equation $\sin(43.833^{\circ})=\frac{d}{13.5}$ by $13.5$. We get $d = 13.5\times\sin(43.833^{\circ})$. Using a calculator, $\sin(43.833^{\circ})\approx0.692$, then $d=13.5\times0.692 = 9.342\approx9.3$ m.

Answer:

$9.3$ m