A crocodile is stalking a prey located 20 metres further upstream on the opposite bank of a river.

It can travel at different speeds on land and in water.

The time taken for the crocodile to reach its prey can be minimized if it swims to a particular point, \(P\), \(x\) metres upstream on the other side of the river as shown in the diagram above.

Let the time taken, \(T\), measured in tenths of a second, be given by

\[ T(x) = 5\sqrt{36+x^2} + 4(20-x) \]

Considering the times \(T\) where \(x= 0\) and \(x =20\), there is a value of \(x\) between 0 and 20 which minimizes \(T\). If \(x=a\) is the point at which \(T(a) \) is minimized, find the value of \(a + T(a) \).

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