The table gives the price, in dollars, of a round$-$trip ticket from Denver to Chicago on a certain airline and the corresponding monthly profit, in millions of dollars, for that airline.

Profit from the Sale of Round$-$Trip Tickets

$($a$)$ Explain why a quadratic model is more appropriate for the data than a log or exponential model.

The data indicates the graph is only decreasing. Logarithmic and exponential functions both increase and decrease while quadratic functions either increase or decrease.

The data indicates a minimum point and changes from decreasing to increasing at that point. Logarithmic and exponential functions do not possess minimum points, but quadratic functions do$.$

The data indicates the graph is only increasing. Logarithmic and exponential functions both increase and decrease while quadratic functions either increase or decrease.

The data indicates a maximum and changes from increasing to decreasing at the point. Logarithmic and exponential functions do not possess maximum points but quadratic functions do$.$

$($b$)$ Find a quadratic model for the data. $($Round all numerical values to six decimal places.$)$

$P(x)$

million dollars

$($c$)$ Why doesn't the airline profit increase as the ticket price increases?

Crew members have their pay increased when ticket prices go up$.$

As ticket prices increase, fewer tickets will be sold at the increased price leading to a decline in profit.

More taxes must be paid on the increased revenue.

The airline has to provide more amenities for passengers to justify the price increase.

$($d$)$ Report the ticket price $($to the nearest dollar$)$ at which the airline will begin to post a negative profit $($that is$,$ a net loss$).$