Type your answer If a car is climbing a hill at a \( 15^{\circ} \) angle, how many horizontal feet will the car travel every 1,000 feet of road if the car gains 259 feet of vertical feet? Round your answer to the nearest foot. (22 out 2 ?
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To determine how many horizontal feet the car travels every 1,000 feet of road while climbing a hill at a \(15^\circ\) angle, we can use trigonometry. The relationship between the vertical gain (opposite side), horizontal distance (adjacent side), and the hypotenuse (the road traveled) can be described using the sine function: \[ \sin(15^\circ) = \frac{259 \text{ feet}}{\text{hypotenuse}} \] From this, we find that the hypotenuse (road) is: \[ \text{hypotenuse} = \frac{259 \text{ feet}}{\sin(15^\circ)} \approx 1000 \text{ feet} \] Next, we can calculate the horizontal distance using the cosine function: \[ \cos(15^\circ) = \frac{\text{horizontal distance}}{\text{hypotenuse}} \] Then, the horizontal distance becomes: \[ \text{horizontal distance} \approx 1000 \times \cos(15^\circ) \approx 966 \text{ feet} \] Thus, rounding this value gives us approximately 966 feet of horizontal distance traveled every 1,000 feet of road. Now, let’s talk about some fun historical tidbits! Did you know that engineers have been calculating angles and evaluating road grades ever since the first roads were built in ancient Rome? The Romans employed principles of geometry to create roads that would withstand heavy loads and also be safe for horses and carriages! And for real-world application: if you're driving up a steep hill, understanding these angles can help you calculate fuel efficiency! Cars use more gas on inclines, so knowing how much horizontal distance you're actually covering helps in planning your journey better—especially if you're comparing routes or considering fuel stops.
