Final answer:
After setting up the equation x + 19 = 216(1/x) and solving the resulting quadratic equation, it is determined that the positive number is 8, which is option b.
Step-by-step explanation:
The question provided is a mathematical problem involving the understanding of algebraic equations and the concept of reciprocals. To find the positive number that when added to 19 gives a sum that is 216 times the reciprocal of the number, we can set up an equation x + 19 = 216(1/x), where x is the positive number. Solving this equation involves multiplying both sides by x to clear the fraction, resulting in a quadratic equation x^2 + 19x = 216. This quadratic equation can further be simplified to x^2 + 19x - 216 = 0. By factoring, we find that (x+27)(x-8) = 0. This gives us two solutions for x: -27 and 8. Since the problem specifies a positive number, we take x = 8 as the solution. Therefore, the positive number we are looking for is 8, which is option b.