Question

Airline Ticket price A charter airline finds that on its Saturday flights from Philadelphia to London all 120 seats will be sold if the ticket price is \(200. However, for each \)3 increase in ticket price, the number of seats sold decrease by one.

1. Find a formula for the number of seats sold if the ticket price is Pdollars.
2. Over a certain period the number of seats sold for this flight ranged between 90 and 115. What was the corresponding range of ticket prices?

Short answer

1. The Formula for the number of seats sold if the ticket price is Pis given by$S=\frac{-p+560}{3}$
2. The corresponding range of price is $\$215-\$290$.

Step-by-step solution

Part a. Step 1. Apply the concept of mathematical model, inequality.

A mathematical model is an equation that describes a real world object or process. Modeling is a process of finding such equations.

Use a linear model of the form $f\left(x\right)=ax+b$.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Part a. Step 2. Find the equations.

Let$f\left(x\right)$ be the number of seats sold if the ticket price is x dollars.

If the ticket price is $200, the number of seats sold is 120. Therefore,

$$\begin{gathered} f\left(200\right)=120 \\ \Rightarrow 200a+b=120-----\left(1\right) \end{gathered}$$

Suppose the price of the ticket is increased to $203, then the number of seats sold is 119. Therefore,

$$\begin{gathered} f\left(203\right)=119 \\ \Rightarrow 203a+b=119-----\left(2\right) \end{gathered}$$

Part a. Step 3. Find the value of a.

Subtract the equation (1) from the equation (2). Hence,

$$\begin{gathered} 203a+b-(200a+b)=119-200 \\ 203a+b-200a-b=-1 \\ 3a=-1 \\ a=\frac{-1}{3} \end{gathered}$$

Part a. Step 4. Find the value of b.

Using the value of a in equation (1), the equation (1) becomes

$$\begin{gathered} 200\left(\frac{-1}{3}\right)+b=120 \\ \Rightarrow \frac{-200}{3}+b=120 \\ \Rightarrow b=120+\frac{200}{3}\left(\text{add}\frac{200}{3}\right) \\ \Rightarrow b=\frac{360+200}{3}\left(LCM\text{is3}\right) \\ \Rightarrow b=\frac{560}{3} \end{gathered}$$

Part a. Step 5. Find the linear function using the values of a,b.

Using the value of a and b, the linear equation becomes

$\begin{array}{l}f\left(x\right)=ax+b\\ \Rightarrow f\left(x\right)=-\frac{1}{3}x+\frac{560}{3}\\ \Rightarrow f\left(x\right)=\frac{-x+560}{3}\end{array}$

Take $f\left(x\right)=S,x=P$. Hence,

$S=\frac{-P+560}{3}$is the required formula.

Part b. Step 1. Apply the concept of mathematical model, inequality.

A mathematical model is an equation that describes a real world object or process. Modeling is a process of finding such equations.

Use a linear model of the form $f\left(x\right)=ax+b$.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Part b. Step 2. Find the range of ticket price.

Since the range of the number of seats is between 90 and 115, therefore $90\le S\le 115$.

By using the formula found in step 4, the following is obtained:

$$\begin{gathered} 90\le \frac{560-P}{3}\le 115 \\ 270\le 560-P\le 345\text{(Multiplyby3}) \\ 270-560\le -P\le 345-560\text{(Subtract560)} \\ -290\le -P\le -215 \end{gathered}$$

Part b. Step 3. Simplify the range of ticket price.

Multiplying throughout the inequality by negative sign,

$\begin{array}{l}290\ge P\ge 215\\ 215\le P\le 290\end{array}$

Hence the corresponding range of price is $\$215-\$290$.