4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladder reaches a 30 -foot roof that needs repair. If the angle of elevation from the bottom of the ladder to the roof is \( 55^{\circ} \), how far is the ladder from the base of the wall? Round your answer to the nearest foot.
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Bonus Knowledge
To find how far the ladder is from the base of the wall, we can use trigonometry. We have the height of the roof (30 feet) and the angle of elevation (55 degrees). Using the sine function: \[ \sin(55^{\circ}) = \frac{opposite}{hypotenuse} = \frac{30}{L} \] where \( L \) is the length of the ladder. Rearranging gives us: \[ L = \frac{30}{\sin(55^{\circ})} \] Now we can find the distance (the adjacent side) from the wall using the cosine function: \[ \cos(55^{\circ}) = \frac{adjacent}{L} \] Thus, the distance from the wall (which we denote as \( d \)) is: \[ d = L \cdot \cos(55^{\circ}) \] Calculating the values gives: 1. \( L \approx 34.88 \) feet (when using a calculator for sine). 2. Then, \( d = 34.88 \cdot \cos(55^{\circ}) \approx 20.00 \) feet. So, the ladder is approximately 20 feet from the base of the wall. Final rounded answer: **20 feet**.
