The length of the tangent can be determined using the concept of the tangent-secant theorem. According to this theorem, if a secant and a tangent are drawn from an external point to a circle, then the product of the lengths of the secant and its external segment is equal to the square of the length of the tangent.

Given:
- Diameter of the circle = 48 m
- Distance of the point P from the center of the circle = 15 cm

Converting the distance of point P from cm to m:
- Distance of point P from the center = 15 cm = 15/100 m = 0.15 m

Finding the length of the tangent:
- The radius of the circle is half the diameter, so the radius = 48/2 = 24 m.
- The distance from the center of the circle to point P is given as 0.15 m.
- The line joining the center of the circle to point P and the tangent drawn from point P form a right-angled triangle.
- The length of the tangent is the hypotenuse of this right-angled triangle.

Applying the Pythagorean theorem:
- According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- In this case, the square of the hypotenuse (length of the tangent) is equal to the sum of the squares of the radius and the distance from the center to point P.

Using the Pythagorean theorem:
- Tangent² = Radius² + Distance from center to point P²
- Tangent² = 24² + 0.15²
- Tangent² = 576 + 0.0225
- Tangent² = 576.0225

Calculating the length of the tangent:
- Taking the square root of both sides, we get:
- Tangent = √(576.0225)
- Tangent ≈ 24.001

Therefore, the length of the tangent is approximately 24.001 m.