First, it's important to ensure that the trigonometric calculations are done correctly. Here are the steps to solve this problem using the Law of Cosines:
1. Convert the angle from degrees to radians:
- The given angle is [tex]\(280^\circ\)[/tex].
- To convert degrees to radians, use the formula: [tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
- So, [tex]\(280^\circ\)[/tex] in radians is:
[tex]\[
280 \times \frac{\pi}{180} \approx 4.88692 \, \text{radians}
\][/tex]
2. Apply the Law of Cosines:
- The Law of Cosines formula is:
[tex]\[
m^2 = a^2 + b^2 - 2ab \cos(\theta)
\][/tex]
- Here [tex]\(a = 27\)[/tex], [tex]\(b = 18\)[/tex], and [tex]\(\theta = 280^\circ\)[/tex] (or [tex]\(4.88692\)[/tex] radians).
- Plugging in the values, we obtain:
[tex]\[
m^2 = 27^2 + 18^2 - 2 \times 27 \times 18 \times \cos(4.88692)
\][/tex]
- Calculate the individual parts:
[tex]\[
27^2 = 729
\][/tex]
[tex]\[
18^2 = 324
\][/tex]
[tex]\[
2 \times 27 \times 18 \approx 972
\][/tex]
[tex]\[
\cos(4.88692) \approx 0.17365
\][/tex]
- Substituting these into the equation:
[tex]\[
m^2 = 729 + 324 - 972 \times 0.17365
\][/tex]
- Simplify the equation:
[tex]\[
m^2 = 1053 - 168.786
\][/tex]
[tex]\[
m^2 \approx 884.214
\][/tex]
3. Calculate the value of [tex]\(m\)[/tex]:
- To find [tex]\(m\)[/tex], take the square root of [tex]\(m^2\)[/tex]:
[tex]\[
m = \sqrt{884.214} \approx 29.7357
\][/tex]
4. Round [tex]\(m\)[/tex] to the nearest whole number:
- Rounding [tex]\(29.7357\)[/tex] to the nearest whole number gives [tex]\(m \approx 30\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{30}
\][/tex]
