Final answer:
To calculate the smallest possible value of z, we set both unknown positive numbers, x and y, to be equal. This simplifies the given expression to 1, as the minimum value possible is when the numerator and denominator are both equal, leading to z = 1 being the answer.
Step-by-step explanation:
In the given problem, we're looking to calculate the smallest possible value of z. This problem involves the mathematical operations of multiplication, addition, and division, as well as the concept of positive numbers and their properties.
Let's denote the unknown positive numbers as x and y. According to the problem, we take x multiplied by itself (x²), then by 4 (4x²), and then add the square of another positive number, y, which gives us (4x² + y²). This sum is then divided by the product of 5, x, and y (5xy), to give us z:
z = (4x² + y²) / (5xy)
Considering that x and y are both positive numbers, the minimum value of this expression is reached when both x and y are equal, i.e., x = y. This is because the square of a number and its multiples are always positive, and they're minimized when x = y. Thus, the equation simplifies to:
z = (4x² + x²) / (5x²) = 5x² / 5x² = 1
Hence, the smallest possible value of z is 1 (Answer B).