To solve for the angle $\theta$ at which the golfer should hit the ball for the ball to travel 588 feet, we are given the formula:

$R(\theta) = 677 \sin(2\theta)$

where:

• $R(\theta)$ is the range in feet,
• $\theta$ is the launch angle,
• and $\sin(2\theta)$ is the sine of double the angle.

We are tasked with finding $\theta$ when $R(\theta) = 588$ feet.

Step-by-Step Solution:

1. Set $R(\theta) = 588$ in the equation: $588 = 677 \sin(2\theta)$

2. Solve for $\sin(2\theta)$: $\sin(2\theta) = \frac{588}{677} = 0.8682$

3. Now, solve for $2\theta$ by taking the inverse sine (arcsin): $2\theta = \arcsin(0.8682)$ Using a calculator: $2\theta \approx 60.99^\circ$

4. Finally, divide by 2 to find $\theta$: $\theta = \frac{60.99}{2} = 30.49^\circ$

Final Answer:

The golfer should hit the ball at an angle of 30.49 degrees.

Would you like any more details, or do you have any other questions?

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Here are 5 additional questions based on this problem:

1. How does the range change if the initial velocity of the ball increases?
2. What would the range be if the golfer hit the ball at a 45-degree angle?
3. How would the range formula change if the initial velocity were different?
4. What is the maximum possible range for this ball, based on the given velocity?
5. What happens to the angle $\theta$ if the target distance increases?

Tip: When solving trigonometric equations, always remember to check if there are multiple solutions for the angle within the required range.