To find the distance the golfer needs to putt to make the shot, let's analyze the situation step-by-step.
1. Identify the Given Values:
- Original distance missed by the putt: 12 feet
- Angle off course: 3°
- Total angle at the hole from the new position: 129°
2. Determine the Effective Angle Between Distances:
- The angle formed by the missed putt, the point where the ball lies now, and the hole is [tex]\( 129° - 3° = 126° \)[/tex].
3. Apply the Law of Cosines to Find the New Distance [tex]\( x \)[/tex]:
- The Law of Cosines is given by:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are sides, and [tex]\( C \)[/tex] is the included angle.
- Here, [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are each 12 feet (the original distance traveled), and [tex]\( C \)[/tex] is 126°.
4. Calculate the New Distance [tex]\( x \)[/tex]:
- Substitute the values into the Law of Cosines:
[tex]\[
x^2 = 12^2 + 12^2 - 2 \times 12 \times 12 \times \cos(126°)
\][/tex]
- Evaluate the cosine part:
[tex]\[
\cos(126°)
\][/tex]
[tex]\[
\cos(126°) \approx -0.5878
\][/tex]
- Plug in and simplify:
[tex]\[
x^2 = 144 + 144 - 2 \times 12 \times 12 \times (-0.5878)
\][/tex]
[tex]\[
x^2 = 144 + 144 + 168.77
\][/tex]
[tex]\[
x^2 \approx 456.77
\][/tex]
[tex]\[
x \approx \sqrt{456.77}
\][/tex]
[tex]\[
x \approx 21.38
\][/tex]
5. Round the Result to the Nearest Hundredth:
- Thus, the distance the golfer needs to putt to make the shot is approximately 21.38 feet.
So, the distance the golfer needs to putt to make the shot is about 21.38 feet.
