Surgery in hospital - linera programming

Hello everyone, I have a problem with this other linear programming problem:

In a large hospital, surgical operations are classified into 3 categories according to their average duration 30 min, 1 hour and 2 hours. The hospital receives a fee of $100,$ 150 and \$ 200 for an operation in categories 1, 2 and 3 respectively. If the hospital has 8 operating rooms which are used an average of 10 hours per day, How many transactions can be programmed so that, a):maximize the incoming money, b) maximizes the number of operations?

SOLUTION:

a) maximize z = 100x1 + 150x2 + 200x3. Restriction $0'5x_1+x_2+2x_3\leq 80$.

Function to maximize $z=x_1+x_2+x_3$, Restriction: $0'5x_1+x_2+2x_3\leq 80,\;x_i\geq{0}$

but my teacher seems to be poorly resolved, because apparently lack an equation ... I do not think most are missing it?

Suddenly, thinking, I thought that the other equation must be of the form
$x_1 + x_2 + x_3 \leq{}$ a given number of operations,

but I am not clear, any idea anyone?

But if that's OK, what would be the dual of this problem?

A greeting and thank you very much