Let's solve the problem step-by-step.

Problem Summary:

1. Initial Conditions:

   • At $8 per ticket, the theater can sell all 1700 seats.
   • For every additional $1 increase in ticket price, 95 fewer tickets are sold.

2. Questions:

   • Part (a): Find the ticket price that maximizes revenue.
   • Part (b): Find the number of seats sold at that price.

Solution:

Let:

• $x$ = number of $1 increases above the base price of $8.
• Ticket price, $P(x) = 8 + x$.
• Tickets sold, $Q(x) = 1700 - 95x$.

Revenue, $R(x)$, is given by: $R(x) = \text{(Ticket Price)} \times \text{(Quantity Sold)} = (8 + x)(1700 - 95x)$

Expanding and simplifying $R(x)$: $R(x) = (8 + x)(1700 - 95x) = 13600 - 760x + 1700x - 95x^2$ $R(x) = -95x^2 + 940x + 13600$

This is a quadratic function in terms of $x$, with a negative leading coefficient, so it opens downward, meaning it has a maximum point.

To find the maximum, we use the vertex formula for a parabola, $x = -\frac{b}{2a}$.

Calculations:

For $R(x) = -95x^2 + 940x + 13600$:

• $a = -95$
• $b = 940$

$x = -\frac{940}{2 \cdot -95} = \frac{940}{190} = 4.947$

Rounding to two decimal places, $x \approx 4.95$.

Substitute $x = 4.95$ to Find Price and Seats:

1. Ticket Price: $P(x) = 8 + 4.95 = 12.95$

   So, the ticket price that maximizes revenue is approximately $12.95.

2. Number of Seats Sold: $Q(x) = 1700 - 95 \cdot 4.95 = 1229.75$

   Rounding to the nearest whole number, 1230 seats are sold at this price.

Answers:

• (a) Ticket price = $12.95
• (b) Number of seats sold = 1230

Would you like more details on any step or have additional questions?

Related Questions:

1. How do we determine if a function opens upwards or downwards in quadratic equations?
2. Why is the revenue maximized at the vertex of the parabola?
3. What happens to the revenue if the price is increased beyond $12.95?
4. How does the ticket price affect the demand in real-life scenarios?
5. Can we solve this problem using calculus methods, and how?

Tip:

In problems involving maximization with a quadratic equation, always check the sign of the leading coefficient—negative means a maximum, while positive means a minimum.