The Apple Pie Cleaning Company receives 405 cases of imported bleach every 27 days. The number of cases of bleach on inventory t days after the shipment arrives is $N(t)=405-15 \sqrt{27 t}$. Find the average daily inventory.

Step 1 ：The Apple Pie Cleaning Company receives 405 cases of imported bleach every 27 days. The number of cases of bleach on inventory t days after the shipment arrives is \(N(t)=405-15 \sqrt{27 t}\). We need to find the average daily inventory.

Step 2 ：The average daily inventory is the total inventory divided by the number of days. In this case, we need to integrate the function \(N(t)\) from 0 to 27 (the number of days in the period) and then divide by 27.

Step 3 ：Let's denote t as the number of days and N as the number of cases. The function for the number of cases is \(N = -45\sqrt{3}\sqrt{t} + 405\).

Step 4 ：By calculating the average inventory, we find that it is 135.

Step 5 ：Final Answer: The average daily inventory is \(\boxed{135}\).