Step 1 :The revenue per day can be calculated using the function:
Step 2 :\( R(x) = \begin{cases} (610 - 7x)(20 + x) & \text{if } 20 < x \leq 90 \\ 12200 & \text{if } x \leq 20 \end{cases} \)
Step 3 :To find the revenue per day when 49 people sign up for the cruise, substitute \( x = 49 - 20 = 29 \) into the function:
Step 4 :\( R(29) = (610 - 7(29))(20 + 29) = 581 \cdot 49 = 28469 \)
Step 5 :Therefore, the revenue per day when 49 people sign up for the cruise is \( \boxed{28469} \).
Step 6 :To find the revenue per day when 84 people sign up for the cruise, substitute \( x = 84 - 20 = 64 \) into the function:
Step 7 :\( R(64) = (610 - 7(64))(20 + 64) = 442 \cdot 84 = 37128 \)
Step 8 :Therefore, the revenue per day when 84 people sign up for the cruise is \( \boxed{37128} \).