To find the angle at which the footballer must kick the ball to score a goal, we can use the Law of Cosines in the context of the triangle formed by the two goalposts and the position of the footballer.
Let:
- \( A \) be the position of the footballer,
- \( B \) be one goal post,
- \( C \) be the other goal post.
We have:
- \( AB = 34 \) m (the distance from the footballer to one post),
- \( AC = 38 \) m (the distance from the footballer to the other post),
- \( BC = 8 \) m (the distance between the two goal posts).
Using the Law of Cosines:
\[ BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(A) \]
Where:
- \( A \) is the angle at the footballer's position.
Plugging in the values:
\[ 8^2 = 34^2 + 38^2 - 2 \cdot 34 \cdot 38 \cdot \cos(A) \]
Calculating each term:
\[ 64 = 1156 + 1444 - 2 \cdot 34 \cdot 38 \cdot \cos(A) \] \[ 64 = 2600 - 2 \cdot 34 \cdot 38 \cdot \cos(A) \]
Now calculate \( 2 \cdot 34 \cdot 38 \):
\[ 2 \cdot 34 \cdot 38 = 2576 \]
So we can rearrange the equation to find \( \cos(A) \):
\[ 64 = 2600 - 2576 \cdot \cos(A) \]
Rearranging gives:
\[ 2576 \cdot \cos(A) = 2600 - 64 \] \[ 2576 \cdot \cos(A) = 2536 \]
Now, solving for \( \cos(A) \):
\[ \cos(A) = \frac{2536}{2576} \] \[ \cos(A) \approx 0.9836 \]
To find the angle \( A \):
\[ A = \cos^{-1}(0.9836) \approx 10.2^\circ \]
Thus, the angle at which the footballer must kick the ball in order to score a goal is approximately 10.2 degrees.
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