Type your answer If a car is climbing a hill at a \( 15^{\circ} \) angle, how many vertical feet will the car travel every 1,000 feet of road? Give your answer rounded to the nearest foot. (Note: \( \sin \left(15^{\circ}\right)=0.259, \cos \left(15^{\circ}\right)=0.966 \), \( \left.\tan \left(15^{\circ}\right) \approx 0.268\right) \)

To find the vertical distance the car travels while climbing the hill, we can use the sine function which relates the angle to the vertical height of a right triangle. The formula is: \[ \text{Vertical Distance} = \text{Distance traveled} \times \sin(\theta) \] Here, the distance traveled is 1,000 feet and \(\theta = 15^{\circ}\): \[ \text{Vertical Distance} = 1000 \times \sin(15^{\circ}) = 1000 \times 0.259 \approx 259 \] To the nearest foot, the car will travel approximately 259 vertical feet. Now let’s get to some fun facts! Did you know that the concept of angles and hills has been a part of navigation and architecture since ancient times? The ancient Greeks contributed significantly to our understanding of geometry and trigonometry, laying the groundwork for how we calculate inclines today! Additionally, this kind of calculation isn’t just for cars! Engineers use similar calculations in designing roads, bridges, and even ski slopes. Understanding the incline helps ensure safety and functionality, making sure drivers and skiers can navigate those slopes with ease. So the next time you’re on a steep hill, remember the science that keeps you safe!