Time, T, is measured in tenths of a second with respect to distance x, is given by T(x)= 5(36+(x^2))^(1/2)+4(20-x). Find the value of x which minimises the time taken, hence calculate the minimum time.

To solve this problem we first must look into the given formula for time (T). It is stated, in the question, that the time taken is dependent on distance x. From this we can infer that the following problem is attempting to find the shortest route between two points. Having stated that, the inputs into the equation are used to estimate the speed required to traverse the various paths. We can see from the equation of T, that two different aspects exists, in this scenario this may mean that 2 different mediums or methods of covering the distance are available.
In order to have the minimum time we know that we have to be travelling at the maximum speed possible. The maximum speed occurs when T'(x) is equal to zero. from this we get the value of x that gives us the minimum time, which we sub back into the time equation to get the minimum time.