Please do not use chatgpt.. please mathematically explain your answer.

Granite State Airlines serves the route between New York City and Portsmouth with

one flight per day on a $100-$seat aircraft. The one$-$way fare for low$-$fare tickets is $$100$ and a one$-$way fare for high$-$fare tickets is $$150.$ Low$-$fare tickets can be booked up until one week in advance and all low$-$fare passengers book before all high$-$fare passengers. The airline estimates that the high$-$fare demand is normally distributed with a mean of $56$ passengers and a standard deviation of $23,$ while low$-$fare$-$fare demand is normally distributed with a mean of $88$ passengers and a standard deviation of $44.$ Normal random variables can take negative

values, but we will ignore this possibility.

$($a$)$ What is the optimal booking limit to impose on the low$-$fare demand? You can directly use the result that we derived in class and keep the booking limit and all demands fractional.

$($b$)$ The airline has been setting a booking limit of $35$ on low$-$fare demand. What is the

expected revenue per flight under this booking limit$?$ Use simulation to answer this question. In Excel, the function $=$norminv$($rand$(),\backslash$mu $,\backslash$sigma $)$ returns a sample from the normal distribution with mean $\backslash$mu and standard deviation $\backslash$sigma $.$ You can use $=$max$($norminv$($rand$(),56,23),0)$ and$=$max$($norminv$($rand$(),88,44),0)$ to generate samples of high$-$fare and low$-$fare demands. Let D$1$ and D$2$ be the samples of low$-$fare and high$-$fare demands. The total revenue with a booking limit of $35$ is given by $100\backslash$times min$\{35,$ D$1\}+150\backslash$times min$\{100$ min$\{35,$ D$1\},$ D$2\}.$ This computation gives one sample of the total revenue corresponding to a given sample of $($D$1,$ D$2).$

You can generate many samples of D$1$ and D$2$ and take the average of the total revenues over many samples. You can place all of the preceding formulas in one row of an Excel spreadsheet and copy these lines for many rows to generate many samples.

$($c$)$ What is the total expected revenue gain from the optimal booking limit over the original booking limit of $35?$ Continue using simulation as in Part $($b$)$ to estimate the total expected revenue from the optimal booking limit