An investor creates a butterfly spread by trading $9-$month call options with strike prices of $$115,$ $$125,$ and $$135.$ The prices of the options are $$20\mathrm{.}50,$ $$14\mathrm{.}50,$ and $$9\mathrm{.}50,$ respectively.

$($Note: Total payoff does not include initial investment$)$

a$.$ What is the initial investment?

b$.$ What is the total payoff when the stock price in $9$ months is $110?$

c$.$ What is the total payoff when the stock price in $9$ months is $120?$

d$.$ What is the total payoff when the stock price in $9$ months is $125?$

e$.$ What is the total payoff when the stock price in $9$ months is $128?$

f$.$ What is the total payoff when the stock price in $9$ months is $140?$