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To solve the problem of finding the shortest path from point (E) to (C_1) on the surface of the rectangular prism, we follow these steps:

Step 1: Key Observations

The rectangular prism has dimensions (AB=10), (AD=6), (AA_1=8). Point (E) is on (AB) with (AE=4), so (EB=10-4=6).

The shortest path on the surface of a prism is found by unfolding adjacent faces into a plane and calculating the straight-line distance between the two points.

Step 2: Unfold Relevant Faces

We consider unfolding the front face ((ABCD)) and the right face ((BCC_1B_1)) into a flat plane:

• The front face ((ABCD)) and right face ((BCC_1B_1)) form a combined rectangle when unfolded along (BC).
• In this plane:
  • (EB=6) (horizontal segment from (E) to (B)),
  • (CC_1=8) (horizontal segment from (C) to (C_1)),
  • (BC=6) (vertical segment connecting the two faces).

Step 3: Calculate Straight-Line Distance

The horizontal distance between (E) and (C_1) in the unfolded plane is (EB + CC_1 = 6 + 8 = 14).
The vertical distance is (BC=6).

Using the Pythagorean theorem:
[EC_1 = \sqrt{14^2 + 6^2} = \sqrt{196 + 36} = \sqrt{232} = 2\sqrt{58}]

Answer: (\boxed{2\sqrt{58}}) (or approximately (\boxed{15}) if rounded, but the exact form is (2\sqrt{58})).

However, if the problem expects an integer, it might be a typo, but the exact shortest path is (\boxed{2\sqrt{58}}).

(\boxed{2\sqrt{58}})

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