A store opens at $7$:$00.$ From $7$:$00$ until $10$:$00,$ customers arrive at a Poisson rate of $20$ an hour. Between $10$:$00$ and $13$:$00$ they arrive at a Poisson rate of $15$ an hour. From $13$:$00$ to $15$:$00$ the arrival rate increases steadily from $15$ per hour at $13$:$00$ to $25$ at $15$:$00.$ From $15$:$00$ to $18$:$00$ the arrival rate drops steadily from $25$ per hour at $15$:$00$ to $19$ per hour at $18$:$00.$ Determine the expected number of customers that enter the store on a given day.