https://r2.erweima.ai/mathbot/image/1725921921003ruxr73l1.jpg

### Problem Breakdown

This problem relates to the pricing of airline tickets and its effect on the number of seats sold.

### Part a:
We are asked to find a formula for the number of seats sold as a function of the price, $ P $.

#### Information given:
- **120 seats** will be sold if the price is **$200**.
- For every increase of **$3**, the number of seats sold decreases by **1**.

We can use the information to model this linear relationship. Let:
- $ S(P) $ be the number of seats sold when the ticket price is $ P $ dollars.

#### Step 1: Write the linear equation
We know two points on the line:
1. $ S(200) = 120 $ (when the price is $200, 120 seats are sold).
2. For every $3 increase in price, the number of seats sold decreases by 1. This gives the slope of the line: 
   \[
   \text{slope} = -\frac{1}{3}
   \]

Thus, the equation for $ S(P) $ is in the form of:
\[
S(P) = mP + b
\]
Where $ m $ is the slope and $ b $ is the y-intercept.

Using the slope $ m = -\frac{1}{3} $ and the point $ (200, 120) $, we can find the equation:

\[
S(P) = -\frac{1}{3}(P - 200) + 120
\]
This simplifies to:
\[
S(P) = -\frac{1}{3}P + \frac{200}{3} + 120
\]
Further simplification will give the formula for the number of seats sold in terms of $ P $.

### Part b:
Now, we are given that during a certain period, the number of seats sold varied between **90** and **115**. We need to determine the corresponding price interval.

This means we need to find the values of $ P $ for which $ S(P) = 90 $ and $ S(P) = 115 $.

#### Step 2: Solve for $ P $
We already have the formula for $ S(P) $, so we will solve:
1. $ S(P) = 90 $ to find the lower price.
2. $ S(P) = 115 $ to find the upper price.

Let me now calculate both parts.### Solution:

#### Part a:
The formula for the number of seats sold, $ S(P) $, in terms of the price $ P $ is:
\[
S(P) = -\frac{1}{3}(P - 200) + 120
\]

#### Part b:
The corresponding prices when the number of seats sold varied between 90 and 115 are:
- For 90 seats: the ticket price was **$290**.
- For 115 seats: the ticket price was **$215**.

Thus, the price interval during this period was **$215 to $290**.

---

Let me know if you'd like more details, or if you have further questions. Here are 5 related questions for deeper exploration:

1. How does the slope of the equation affect the number of seats sold?
2. What would happen to the number of seats sold if the price were lowered to $180?
3. Can you derive the equation for the price $ P $ if the number of seats sold is known?
4. What other factors could affect the relationship between ticket prices and seats sold?
5. How would the graph of this function look, and where would the key points be?

### Tip:
Always check the units in real-world problems like this to ensure your results are consistent.

Una aerolínea que realiza vuelos especiales encuentra que para los vuelos de los sábados, de Filadelfia a Londres, los 120 asientos se venderán si el precio es de 200 dólares. No obstante, por cada aumento de 3 dólares en el precio del boleto, el número de asientos disminuye en uno. Encuentre una fórmula para el número de asientos vendidos si el precio del boleto es P dólares. Durante cierto período el número de asientos vendidos para este vuelo variaba entre 90 y 115. ¿Cuál era el intervalo correspondiente de precios de los boletos?

Solve a linear function problem where the number of seats sold decreases as the price of airline tickets increases. Given initial conditions, find a formula for seat sales and determine the price range for varying seat sales. Suitable for high school students learning about linear functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation