Question: A small cruising ship that can hold up to 66 people provides three-day excursions to groups of 44 or more. If the group contains 44 people, each person pays 78. The cost per person for all members of the party is reduced by 1 for each person in excess of 44. Find the size of the group that maximizes income for the owners of the ship.
1. **State the problem:** We want to find the group size $x$ (where $44 \leq x \leq 66$) that maximizes the total income for the ship owners.
2. **Define variables:** Let $x$ be the number of people in the group.
3. **Write the price per person:** When there are 44 people, price per person is $78$. For each person above 44, price decreases by $1$. So price per person is:
$$p(x) = 78 - (x - 44) = 122 - x$$
4. **Write the income function:** Total income $I(x)$ is number of people times price per person:
$$I(x) = x \times p(x) = x(122 - x) = 122x - x^2$$
5. **Maximize the income:** This is a quadratic function $I(x) = -x^2 + 122x$ which opens downward, so maximum is at vertex.
6. **Find vertex:** The vertex $x$-coordinate is given by:
$$x = -\frac{b}{2a} = -\frac{122}{2 \times (-1)} = \frac{122}{2} = 61$$
7. **Check domain:** $61$ is between $44$ and $66$, so valid.
8. **Conclusion:** The group size that maximizes income is $\boxed{61}$ people.