To solve for the length of the diagonal [tex]\(x\)[/tex] in the given parallelogram using the law of cosines, we can follow these steps:
1. Identify the given values:
- The side lengths of the parallelogram are [tex]\( a = 4 \)[/tex] and [tex]\( b = 6 \)[/tex].
- The angle between these two sides is [tex]\( A = 55^{\circ} \)[/tex].
2. Convert the angle from degrees to radians:
- The angle [tex]\( A \)[/tex] in degrees is [tex]\( 55^{\circ} \)[/tex].
- Converting degrees to radians gives [tex]\( \approx 0.9599 \)[/tex] radians.
3. Apply the law of cosines:
[tex]\[
x^2 = a^2 + b^2 - 2ab \cos(A)
\][/tex]
- Substitute the known values into the formula:
[tex]\[
x^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos(55^{\circ})
\][/tex]
- First, calculate each part:
- [tex]\( 4^2 = 16 \)[/tex]
- [tex]\( 6^2 = 36 \)[/tex]
- The cosine of [tex]\( 55^{\circ} \)[/tex] is approximately [tex]\( \cos(55^{\circ}) \approx 0.5736 \)[/tex]
- [tex]\( 2 \cdot 4 \cdot 6 \cdot 0.5736 \approx 27.59 \)[/tex]
- So,
[tex]\[
x^2 = 16 + 36 - 27.59 \approx 24.41
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Take the square root of both sides to find [tex]\( x \)[/tex]:
[tex]\[
x \approx \sqrt{24.41} \approx 4.95
\][/tex]
5. Round to the nearest whole number:
- The decimal value [tex]\( 4.95 \)[/tex] rounds to [tex]\( 5 \)[/tex].
After following these steps, we find that the length of the diagonal [tex]\( x \)[/tex] to the nearest whole number is [tex]\( \boxed{5} \)[/tex].
