A golfer hits a golf ball with an initial velocity of 102 miles per hour. The range R of the ball as a function of the angle 0 to the horizontal is given by R(teta) = 677 sin (20), where R is measured in feet Complete
parts (a) through (c).

(a) At what angle teta should the ball be hit if the golfer wants the ball to travel 588 feet (196 yards)?

Teta= …. Degree

(Type an integer or decimal rounded to the nearest hundredth as needed.)

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?

---

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.

A golfer hits a golf ball with an initial velocity of 102 miles per hour. The range R of the ball as a function of the angle 0 to the horizontal is given by R(teta) = 677 sin (2teta), where R is measured in feet. At what angle teta should the ball be hit if the golfer wants the ball to travel 588 feet (196 yards)?

Learn how to calculate the angle at which a golfer should hit a golf ball with an initial velocity of 102 mph to reach a distance of 588 feet. This problem involves trigonometric functions and projectile motion, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation