Being the 90th percentile means to be better than 90% of the dataset. Since this dataset is composed of 10 elements, the 90th percentile will be exactly the 9th element.
So, you sort your dataset to get
[tex] 5581, 5700, 5700, 5896, 5972, 5993, 6075, 6274, 6283, 6381 [/tex]
which means that the 90th percentile is 6283
