Consider equation (1)
Now, let us complete the squares on the right hand side of equation (5).
If we look at the right hand side of equation (6), the first term 0.03(t-13/2)2 is either zero or positive for any t value and the second term is positive (check this) all the time. This means that W will have a minimum when the first term is zero (are you clear on this?)
i.e. W will have a minimum when
and this is exactly the same answer we got earlier, in equation (4), by using the derivative approach!