X, Y, and Z can complete the work individually in 6 hours, 8 hours, and 8 hours, respectively. To minimize the total time required to finish the work, we need to utilize the most efficient worker, X, as much as possible. Thus, we will alternate between X and either Y or Z, starting with X.

Work Rates:

• X completes the work in 6 hours, so in one hour, X completes $1/6$ of the work.
• Y (or Z) completes the work in 8 hours, so in one hour, Y (or Z) completes $1/8$ of the work.

Work Distribution: We will alternate between X and Y (or Z) in the following pattern:

• First hour: X works, completing $1/6$ of the work.
• Second hour: Y (or Z) works, completing $1/8$ of the work.
• Third hour: X works, completing $1/6$ of the work.
• Fourth hour: Y (or Z) works, completing $1/8$ of the work.
• Fifth hour: X works, completing $1/6$ of the work.

After 6 hours, the total work completed is:

• 3X + 3Y = $3 \times 1/6 + 3 \times 1/8 = 1/2 + 3/8 = 7/8$

Remaining Work: After 6 hours, $7/8$ of the work is completed, leaving $1/8$ of the work remaining. Since it’s now X’s turn to work, we calculate how much time X will need to complete the remaining $1/8$ of the work:

• X completes $1/6$ of the work in 1 hour,
• So to complete $1/8$, the time required is: $(1/8 \div 1/6)$ hours = $3/4$ hours = $3/4 \times 60$ minutes = 45 minutes

Total Time: Thus, the total time to complete the work is:

• 6 hours (for the first $7/8$ of the work) + 45 minutes (for the remaining $1/8$) = 6 hours 45 minutes.

Thus, the correct answer is C. 6 hours 45 minutes.