To solve for the angle \( x \) using the Law of Cosines, we will determine which angle we want to calculate. Let's assume that we want to find the angle opposite side \( c \). The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(x) \]
Where:
- \( a = 5.0 \)
- \( b = 7.7 \)
- \( c = 9.2 \)
Substituting in the values we have:
\[ (9.2)^2 = (5.0)^2 + (7.7)^2 - 2(5.0)(7.7) \cdot \cos(x) \]
Calculating each term:
\[ 84.64 = 25 + 59.29 - 77 \cdot \cos(x) \]
Now simplify:
\[ 84.64 = 84.29 - 77 \cdot \cos(x) \]
Rearranging gives:
\[ 84.64 - 84.29 = -77 \cdot \cos(x) \]
\[ 0.35 = -77 \cdot \cos(x) \]
Now isolate \( \cos(x) \):
\[ \cos(x) = -\frac{0.35}{77} \]
Calculating \( \cos(x) \):
\[ \cos(x) = -0.00454545 \]
Now, to find \( x \), we take the inverse cosine:
\[ x = \cos^{-1}(-0.00454545) \]
Calculating \( x \):
\[ x \approx 90.3^\circ \text{ (taking into account the negative value results in an angle over 90 degrees)} \]
Thus, rounding to the nearest tenth, the answer is:
x is equal to 90.3 degrees.