To solve for [tex]\( 2ab \cos C \)[/tex] using the given law of cosines, we start with the formula:
[tex]\[ a^2 + b^2 - 2ab \cos C = c^2 \][/tex]
Our goal is to isolate [tex]\( 2ab \cos C \)[/tex]. We can do this by rearranging the formula:
[tex]\[ a^2 + b^2 - c^2 = 2ab \cos C \][/tex]
Now, noticing that the expression [tex]\( a^2 + b^2 - c^2 \)[/tex] is equal to [tex]\( 2ab \cos C \)[/tex], we'll substitute the given values for [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
Let's assume:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Substituting these values into the equation:
[tex]\[ 2ab \cos C = 5^2 + 8^2 - 7^2 \][/tex]
[tex]\[ 2ab \cos C = 25 + 64 - 49 \][/tex]
[tex]\[ 2ab \cos C = 40 \][/tex]
Therefore, the value of [tex]\( 2ab \cos C \)[/tex] is:
[tex]\[ \boxed{40} \][/tex]
So, the correct answer is [tex]\( C. 40 \)[/tex].
