Hello S N,
Let's let $C$ be the amount charged in dollars and $P$ be the number of people that attend.
We are told that we $P$ increases by 10, then $C$ decreases by 1.5, so we may state the slope of the line is:
$$m=\frac{\Delta P}{\Delta C}=\frac{10}{-1.5}=-\frac{20}{3}$$
we are given the point on the line $(30,120)$, and so using the point-slope formula, we may determine the linear relationship between $P$ and $C$ as:
$$P-120=-\frac{20}{3}(C-30)$$
which we may arrange in slope-intercept form as:
$$P(C)=-\frac{20}{3}C+320$$
Now, the revenue $R$ for the hall is the product of the charge per person times the number of people attending, hence:
$$R(C)=C\cdot P(C)=C\left(-\frac{20}{3}C+320 \right)=-\frac{20}{3}C(C-48)$$
We know the vertex of this parabolic revenue function will be on the axis of symmetry, which will be midway between the two roots, at $C=0,\,48$, which means the axis of symmetry is the line $C=\dfrac{0+48}{2}=24$.
Thus, revenue is maximized when the number of people attending is given by:
$$P(24)=-\frac{20}{3}\cdot24+320=-160+320=160$$
Thus, when 160 people attend, revenue is maximized.
