Step 1: Understand the Proportion

The given expression $48: x:: x: 75$ means that $48$, $x$, and $75$ are in continued proportion.

$\frac{48}{x} = \frac{x}{75}$

Step 2: Cross-Multiply to Solve for $x^2$

Cross-multiplying the terms gives $x^2 = 48 \times 75$.

$x^2 = 3600$

Step 3: Extract the Square Root

Taking the square root of both sides, and given that $x > 0$, we find the positive value of $x$.

$x = \sqrt{3600} = 60$