Two paths lead to the top of a big hill. One is steep and direct, while the other is twice as long but less steep. How much more gravitational potential energy would you gain if you took the longer path

None.

Explanation:

Gravitational potential energy, depends only of the mass on which the gravity is doing work, and the displacement produced by this force.As displacement depends only on the final and initial positions (in this case, the height of the hill), if we choose as our zero reference level the bottom of the hill, the change in gravitational potential energy will be as follows:

       

As we can see, the only value of distance involved is the height of the hill , so it is independent of the distance travelled.

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