analyze the following situation, and explain what decision you would make and why. you fly frequently…

Explanation:

Step1: Calculate number of round - trips for free ticket

The distance for a round - trip is 1500 miles. To earn a free round - trip on Airline A (after 24000 miles), the number of round - trips $n=\frac{24000}{1500}=16$.

Step2: Set up cost functions

Let $x$ be the number of round - trips. The cost function for Airline A, $C_A = 325x$ (before getting the free ticket). After getting the free ticket (when $x\geq16$), the effective cost is for $x - 1$ trips, so $C_A=325(x - 1)$ for $x\geq16$. The cost function for Airline B is $C_B = 300x$.

Step3: Compare costs for $x < 16$

For $x < 16$, we compare $C_A$ and $C_B$. Let's find when $C_A<C_B$, $325x<300x$ gives $25x < 0$ (no solution). Let's find when $C_A = C_B$, $325x=300x$ gives $25x = 0$ (only solution is $x = 0$). Let's find when $C_A>C_B$, $325x>300x$ which is true for $x>0$. So for $x < 16$, Airline B is cheaper.

Step4: Compare costs for $x\geq16$

For $x\geq16$, $C_A = 325(x - 1)=325x-325$ and $C_B = 300x$. Set $C_A<C_B$, $325x-325<300x$, $325x-300x<325$, $25x<325$, $x < 13$ (contradiction since we are considering $x\geq16$). Set $C_A = C_B$, $325x-325 = 300x$, $25x=325$, $x = 13$ (contradiction). Set $C_A>C_B$, $325x-325>300x$, $25x>325$, $x>13$. So for $x\geq16$, Airline A is cheaper.

Answer:

A. Flying round - trip under 17 times, Airline B is cheaper. Flying round - trip 17 times or more, Airline A is cheaper.