Hey guys, ok i'll start off with my answer! PS: I have no idea how to make my written answers show appropriately.
For the quadratic model, I get:
f(x)=1/80x2+15
and for the logarithmic model:
f(x)=30(1+ Ln(x/25)/Ln(12/5))
------------Re-checking ---------------------
f(x)=30+30ln(125)ln(x25)
=30+30ln(125)ln(x)−30ln(25)ln(125)
=30⎛⎝⎜⎜1−ln(25)ln(125)⎞⎠⎟⎟+30ln(125)ln(x)
a=30⎛⎝⎜⎜1−ln(25)ln(125)⎞⎠⎟⎟≈−80.302
b=30ln(125)≈34.267
**********
Solving the linear system:
a+bln(25)=30
a+bln(60)=60
b(ln(60)−ln(25))=30
b=30ln(60)−ln(25)=30ln(125)
Substituting this in the first equation:
a=30−bln(25)=30−⎛⎝⎜⎜30ln(125)⎞⎠⎟⎟ln(25)
=30⎛⎝⎜⎜1−ln(25)ln(125)⎞⎠⎟⎟
I didn't want to post up the entire question below because I'm pretty much almost done with it. Im having trouble finding the quadratic and logarithmic models with NO INCLUDED gigabytes, i have no idea how to get the equations to put them in a graph, help would be appreciated. (the above answer is for the normal graph, with the included gigabytes).
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an internet provider is reviewing its monthly broadband usage packs, currently, customers can choose one of the three following options:
CURRENT PACKS:
appetizer : monthly access fee $7, included gigabyte 10, cost per additional gigabyte $2
quest: monthly access fee $40, included gigabyte 40, cost per additional gigabyte $1
voyager: monthly access fee $60, included gigabyte 60, cost per additional gigabyte $0.50
these packs comprise a monthly access fee which includes usage up to a certain number of gigabytes plus an additional cost per gigabyte over and above that.
The internet provider intends to discontinue the quest pack. they are considering two new pack options to replace it.
Replacement pack options:
(option 1) quadratic model, monthly access fee $20, included gigabytes 20, cost per additional gigabyte *see below (*)
(option 2) logarithmic model, monthly access fee $30, included gigabytes 25, cost per additional gigabyte *see below (**)
(*) The charge option 1 will increase quadratically after 20 gigabytes with customers paying the same amount as voyager customers at 60 gigabytes.
(**) The charge option 2 will increase logarithmically in the form ƒ(x) = a + b 1n|x| after 25 gigabytes with customers also paying the same amount as voyager customers at 60 gigabytes.
in addition, they are also looking into the possibility of removing the 'included gigabyte usage' from each pack . this would mean that the customer would pay the monthly access fee plus the cost of each gigabyte used.