1. **State the problem:** We want to find the ticket price $t$ that maximizes the profit $P$ given by the quadratic relation $$P = -37t^2 + 1258t - 7700.$$ 2. **Recall the formula and method:** A quadratic function in standard form $$y = ax^2 + bx + c$$ can be rewritten in vertex form $$y = a(x - h)^2 + k$$ by completing the square. The vertex $(h,k)$ gives the maximum or minimum value depending on the sign of $a$. Since $a < 0$ here, the vertex will give the maximum profit. 3. **Complete the square:** Start with $$P = -37t^2 + 1258t - 7700.$$ Factor out $-37$ from the first two terms: $$P = -37(t^2 - \frac{1258}{37}t) - 7700.$$ Calculate $\frac{1258}{37} = 34$, so $$P = -37(t^2 - 34t) - 7700.$$ 4. **Create a perfect square:** Take half of 34, which is 17, and square it: $17^2 = 289$. Add and subtract 289 inside the parentheses: $$P = -37(t^2 - 34t + 289 - 289) - 7700.$$ Group the perfect square trinomial: $$P = -37((t - 17)^2 - 289) - 7700.$$ 5. **Distribute and simplify:** $$P = -37(t - 17)^2 + 37 \times 289 - 7700.$$ Calculate $37 \times 289 = 10703$, so $$P = -37(t - 17)^2 + 10703 - 7700.$$ Simplify the constants: $$P = -37(t - 17)^2 + 3003.$$ 6. **Interpret the vertex form:** The vertex is at $t = 17$ and the maximum profit is $P = 3003$. **Final answer:** The ticket price that maximizes profit is $\boxed{17}$, and the maximum expected profit is $\boxed{3003}$.