You need some protein after work and before that workout. In fact, make sure you are getting a protein snack regularly throughout the day (I work in a school district and carry low-fat string cheese, protein bars, and Wasa and Laughing Cow cheese with me every day).
When I finish a TOUGH day in the district, I try to have a large glass of water AND a protein snack. By the time I reach home, I am ready to tackle my workout. I usually grab another protein snack (nuts are my snack of choice) and more water when I finish.
You CAN do this - but be smart about it! Carrots are a GREAT snack, but not enough "fuel" for your workout!