Let's set up the given problem as an equation and solve for x.

The product of the number 7 and a number x is divided by 4:

(7x) / 4

The square root of the resulting number is twice x:

√((7x) / 4) = 2x

We can square both sides of the equation to eliminate the square root:

(√((7x) / 4))^2 = (2x)^2

Simplifying further:

(7x) / 4 = 4x^2

Multiplying both sides of the equation by 4 to eliminate the fraction:

7x = 16x^2

Bringing all terms to one side of the equation:

16x^2 - 7x = 0

Now we can factor out x:

x(16x - 7) = 0

Setting each factor to zero separately:

x = 0 or 16x - 7 = 0

If x = 0, it does not satisfy the condition x > 0 given in the problem.

So we focus on the second factor:

16x - 7 = 0

Adding 7 to both sides:

16x = 7

Dividing both sides by 16:

x = 7/16

So the value of x that satisfies the given conditions (x > 0) is x = 7/16. Therefore, the correct option is "7/16".