Suppose Alex moves a distance x km on sand from starting point to the road and from road to base camp.

The distance traveled on the road will be 45 - 2*sqrt[(x^2) - (10)^2]

Time taken by him on the sand = 2x/30 h.

Time taken by him on the on the road = { 45 - 2*sqrt[(x^2) - (10)^2]}/50

Find the total time T.

To find the minimum distance traveled on sand, find dT/dx and equate it to zero. Once you find x, you can find the minimum time taken by Alex to reach the base camp.