Given information:
- The mean of four numbers is 37.
- The mean of the smallest three of them is 34.
- The range of the data is 15.

To find: The mean of the largest three numbers.

Let's consider the four numbers as a, b, c, and d, where a ≤ b ≤ c ≤ d.

Finding the Mean of the Four Numbers:
The mean of the four numbers is given as 37. The mean is the sum of all the numbers divided by the total number of values. Therefore, we can write the equation as:
(a + b + c + d) / 4 = 37

Finding the Mean of the Smallest Three Numbers:
The mean of the smallest three numbers is given as 34. Since a, b, and c are the smallest three numbers, we can write the equation as:
(a + b + c) / 3 = 34

Finding the Range:
The range is the difference between the largest and smallest numbers. Here, the range is given as 15. Therefore, we can write the equation as:
d - a = 15

Solving the Equations:
We have three equations with three unknowns (a, b, c). We can solve them simultaneously to find the values of a, b, c, and d.

From equation 2, we can express a in terms of b and c:
a = 102 - b - c

Substituting a in equation 3, we have:
(102 - b - c) + b + 15 = 15
102 - c = 15
c = 87

Substituting the value of c in equation 2, we have:
a + b + 87 = 102
a + b = 15

Substituting the values of a and c in equation 1, we have:
(15 + b + 87 + d) / 4 = 37
(102 + b + d) / 4 = 37
102 + b + d = 148
b + d = 46

Solving the equations b + d = 46 and a + b = 15, we find:
a = 0, b = 15, c = 87, d = 31

Therefore, the numbers are 0, 15, 87, and 31.

Mean of the Largest Three Numbers:
To find the mean of the largest three numbers (b, c, d), we can calculate the sum of these numbers and divide it by 3:
Mean = (15 + 87 + 31) / 3
Mean = 133 / 3
Mean ≈ 44.33

Therefore, the mean of the largest three numbers is approximately 44.33, which is closest to option D) 39.