A store opens at $8$:$00$ and closes at $18$:$00.$ From $8$:$00$ to $11$:$00,$ customers arrive at a Poisson rate of five per hour. Between $11$:$00$ and $13$:$00,$ customers arrive at a Poisson rate of six per hour. From $13$:$00$ to $16$:$00,$ the arrival rate increases steadily from six per hour at $13$:$00$ to $12$ per hour at $16$:$00.$ From $16$:$00$ to $18$:$00,$ the arrival rate drops steadily from $12$ per hour at $16$:$00$ to six per hour at $18$:$00.$

$(5\mathrm{.}1)$ Briefly explain why this process is a nonhomogeneous Poisson process.

$(5\mathrm{.}2)$ Write down the intensity function of the process.

$(5\mathrm{.}3)$ Calculate the mean of the daily arrivals.