Pregunta
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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.

Ask by Herrera Gross. in the United States
Mar 20,2025

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The ladder is approximately 21 feet away from the base of the wall.

Solución

**Step 1: Identify the right triangle components** The ladder, wall, and ground form a right triangle. The height of the roof is the side opposite the \(55^\circ\) angle, and the distance from the ladder to the wall is the side adjacent to the angle. **Step 2: Use the tangent function** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, \[ \tan(55^\circ) = \frac{30}{x} \] where \(x\) is the horizontal distance from the base of the wall to the ladder. **Step 3: Solve for \(x\)** Rearrange the equation to solve for \(x\): \[ x = \frac{30}{\tan(55^\circ)} \] **Step 4: Calculate \(x\)** First, compute \(\tan(55^\circ)\). Using a calculator, \[ \tan(55^\circ) \approx 1.4281 \] Then, \[ x \approx \frac{30}{1.4281} \approx 21 \] **Step 5: Round the answer** Since the problem asks for the answer rounded to the nearest foot, \[ x \approx 21 \text{ feet} \] **Final Answer:** \(21\) feet.

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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**.

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