To find the value of [tex]\( m \)[/tex] using the Law of Cosines, let's follow the detailed solution step-by-step:
1. We start with the initial given values:
[tex]\[
a = 27, \quad b = 18, \quad \text{and} \quad \angle = 28^\circ
\][/tex]
2. Convert the angle from degrees to radians. (Note: This step is necessary because trigonometric functions in calculations typically require the angle in radians, but this conversion detail is not reflected in your Law of Cosines formula directly.)
3. Apply the Law of Cosines formula:
[tex]\[
m^2 = a^2 + b^2 - 2ab \cos(\angle)
\][/tex]
Plugging in the values, we have:
[tex]\[
m^2 = 27^2 + 18^2 - 2 \cdot 27 \cdot 18 \cdot \cos(28^\circ)
\][/tex]
4. Calculate each term separately:
[tex]\[
27^2 = 729
\][/tex]
[tex]\[
18^2 = 324
\][/tex]
[tex]\[
2 \cdot 27 \cdot 18 = 972
\][/tex]
[tex]\[
\cos(28^\circ) \approx 0.8829475928589269 \quad (\text{using a calculator to find the cosine})
\][/tex]
5. Substitute these values back into the equation:
[tex]\[
m^2 = 729 + 324 - 972 \cdot 0.8829475928589269
\][/tex]
6. Multiply:
[tex]\[
972 \cdot 0.8829475928589269 \approx 858.2250602588771
\][/tex]
7. Simplify:
[tex]\[
m^2 = 729 + 324 - 858.2250602588771
\][/tex]
[tex]\[
m^2 = 1053 - 858.2250602588771
\][/tex]
[tex]\[
m^2 \approx 194.77493974112292
\][/tex]
8. Take the square root to find [tex]\( m \)[/tex]:
[tex]\[
m \approx \sqrt{194.77493974112292} \approx 13.95617926730389
\][/tex]
9. Round [tex]\( m \)[/tex] to the nearest whole number:
[tex]\[
m \approx 14
\][/tex]
Thus, the value of [tex]\( m \)[/tex] to the nearest whole number is:
[tex]\[
\boxed{14}
\][/tex]
