To solve for the value of [tex]\( m \)[/tex] using the Law of Cosines, we follow these steps:
1. State the Law of Cosines formula:
[tex]\[
m^2 = a^2 + b^2 - 2ab \cos(\theta)
\][/tex]
Here, [tex]\( a = 27 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( \theta = 28^\circ \)[/tex].
2. Substitute the given values into the formula:
[tex]\[
m^2 = 27^2 + 18^2 - 2 \cdot 27 \cdot 18 \cdot \cos(28^\circ)
\][/tex]
3. Calculate the squares of the sides:
[tex]\[
27^2 = 729
\][/tex]
[tex]\[
18^2 = 324
\][/tex]
4. Sum the squares:
[tex]\[
729 + 324 = 1053
\][/tex]
5. Find the cosine of [tex]\( 28^\circ \)[/tex] (using a calculator or a trigonometric table, but let's just consider the given result):
[tex]\[
\cos(28^\circ) \approx 0.882947593
\][/tex]
6. Calculate the product [tex]\( 2 \cdot 27 \cdot 18 \cdot \cos(28^\circ) \)[/tex]:
[tex]\[
2 \cdot 27 \cdot 18 \cdot 0.882947593 = 972 \cdot 0.882947593 \approx 858.225060258877
\][/tex]
7. Subtract this product from the sum of the squares:
[tex]\[
m^2 = 1053 - 858.225060258877 \approx 194.77493974112292
\][/tex]
8. Take the square root of [tex]\( m^2 \)[/tex] to find [tex]\( m \)[/tex]:
[tex]\[
m = \sqrt{194.77493974112292} \approx 13.95617926730389
\][/tex]
9. Round [tex]\( m \)[/tex] to the nearest whole number:
[tex]\[
m \approx 14
\][/tex]
Therefore, to the nearest whole number, the value of [tex]\( m \)[/tex] is [tex]\( 14 \)[/tex]. The correct answer is:
[tex]\[
\boxed{14}
\][/tex]
