Problem Analysis:
We are given that there are 50 integers written in non-decreasing order and their average is 15. We are also given that the 30th integer is 30. We need to find the highest possible value of the average of the first 30 integers.

Solution:
To find the highest possible value of the average of the first 30 integers, we need to maximize the sum of the first 30 integers while keeping the average at 15.

Step 1: Finding the sum of the first 30 integers:
Since the average of the 50 integers is 15, the sum of all the 50 integers can be found by multiplying the average by the total number of integers:
Sum of all integers = Average * Total number of integers
Sum of all integers = 15 * 50 = 750

Now, we need to find the sum of the last 20 integers. Since the 30th integer is 30, all the integers from the 31st to the 50th will be greater than or equal to 30. We can assume the remaining 20 integers as 30, so their sum will be 20 * 30 = 600.

The sum of the first 30 integers can be found by subtracting the sum of the last 20 integers from the sum of all the integers:
Sum of the first 30 integers = Sum of all integers - Sum of the last 20 integers
Sum of the first 30 integers = 750 - 600 = 150

Step 2: Finding the highest possible value of the average:
To find the highest possible value of the average of the first 30 integers, we divide the sum of the first 30 integers by the total number of integers:
Average of the first 30 integers = Sum of the first 30 integers / Total number of integers
Average of the first 30 integers = 150 / 30 = 5

Therefore, the highest possible value of the average of the first 30 integers is 5.

Conclusion:
The highest possible value of the average of the first 30 integers is 5. This can be achieved by assuming the remaining 20 integers as 30 and subtracting their sum from the sum of all the integers.