Question

A golfer hits a golf ball with an initial velocity of 100 miles per hour. the range r of the ball as a function of the angle θ to the horizontal is given by r(θ)=670sin(2θ)​, where r is measured in feet. complete parts​ (a) through​ (c) below. question content area bottom part 1 ​(a) at what angle θ should the ball be hit if the golfer wants the ball to travel 447 feet ​(149 ​yards)?

Answer 1

To find the angle θ for a 447-foot golf shot, the equation 447 = 670sin(2θ) is used. After isolating sin(2θ) and using the inverse sine function, θ is found by dividing the result by two. The projectile's range is notably affected by initial velocity and projection angle.

Step-by-step explanation:

To determine at what angle θ the ball should be hit for the golfer to achieve a range of 447 feet, we can use the given range function r(θ)=670sin(2θ). Setting r(θ) to 447 feet and solving for θ will provide the answer:

1. Start with the equation 447 = 670sin(2θ).
2. Divide both sides by 670 to isolate the sin(2θ) term, getting approximately sin(2θ) = 0.667.
3. Use the inverse sine function to find 2θ, then divide by 2 to solve for θ. Remembering that the inverse sine function can yield multiple results due to the periodic nature of the sine function, one should consider possible angles in the range where a projectile would realistically be launched.

Understanding that the initial velocity and projection angle profoundly affect the range of a projectile in physics is crucial. For example, setting the angle at 45° maximizes the range under ideal conditions without air resistance, but if air resistance is considered, the maximum angle becomes approximately 38°. This knowledge is helpful when solving real-world problems such as this golfing scenario.