I have problems answering questions like this....(i'll just make up a question)

The time it takes to bake a cake in a bakery shop is a random variable that has a normal distribution with a mean of 4.5 minutes and standard deviation of 1 minute.

Lets suppose this bakery has been given an order to make 78 cakes. What is the probability that the baker will have in less than 380 minutes all 78 cakes ready?

My first instinct is to treat this as a poisson distribution but if I was to evaulate P[N < 79] it would simply take too long.

Any ideas would be great :)