Let's assume that the number of students in the class is x.
The formula for the instructor's earnings would be:
Earnings = Flat rate + (Per student rate * Number of students)
Earnings = $100 + ($32 * x)
We want to determine the number of students needed for the earnings to be $516, so we can set up an equation:
$516 = $100 + ($32 * x)
Now we can solve for x:
$416 = ($32 * x)
x = 13
Therefore, the instructor would need 13 students to enroll in the pottery class for their earnings to be $516.