Quote (NomadJe @ Jul 7 2023 09:38am)

Part A
It depends if we consider a general function on real numbers or a realistic area function (where, for example, negative numbers make no sense).
In the first case: Dom{A} = {r in R / A(r) is defined on R} = R, because for each value of radius there is a corresponding A(r) (it's a basic parabolic function)
In the second case: Dom{A} = {r in R+ / A(r) in R+} = R+, because for each positive value of radius there is a corresponding A(r) that is positive

Part B
You have to find the root solutions of 2*pi*r^2 + 16*pi*r - A = 0
This leads to r(A) = -4 +/- sqrt(16 + A/2pi)
Since this is not a function because "+/-" does not lead to a unique value, we assume for simplicity the second case of Part A, so that both A and r must be positive to make sense.
This means r(A) = -4 + sqrt(16 + A/2pi) because r(A) -4 - sqrt(etc...) always lead to a negative number
16 + A/2pi >= 0 because of square root --> A>= -32pi (always true, we assumed A>=0)
Moreover, if A=0 then r(A)=0 which makes sense (a cylinder with 0 radius has 0 area)

Part C
This is just r(250) = -4 + sqrt(16 + 250/2pi) = -4 + sqrt(55,79) = 3,47
Lets verify this, A(3,47) = 2*pi*(12,04) + 16*pi*3,47 = 75,65 + 174,42 = 250,07 (0,07 of mistake due some rounding up)

Quote (NomadJe @ Jul 7 2023 09:38am)

Part A
It depends if we consider a general function on real numbers or a realistic area function (where, for example, negative numbers make no sense).
In the first case: Dom{A} = {r in R / A(r) is defined on R} = R, because for each value of radius there is a corresponding A(r) (it's a basic parabolic function)
In the second case: Dom{A} = {r in R+ / A(r) in R+} = R+, because for each positive value of radius there is a corresponding A(r) that is positive

Part B
You have to find the root solutions of 2*pi*r^2 + 16*pi*r - A = 0
This leads to r(A) = -4 +/- sqrt(16 + A/2pi)
Since this is not a function because "+/-" does not lead to a unique value, we assume for simplicity the second case of Part A, so that both A and r must be positive to make sense.
This means r(A) = -4 + sqrt(16 + A/2pi) because r(A) -4 - sqrt(etc...) always lead to a negative number
16 + A/2pi >= 0 because of square root --> A>= -32pi (always true, we assumed A>=0)
Moreover, if A=0 then r(A)=0 which makes sense (a cylinder with 0 radius has 0 area)

Part C
This is just r(250) = -4 + sqrt(16 + 250/2pi) = -4 + sqrt(55,79) = 3,47
Lets verify this, A(3,47) = 2*pi*(12,04) + 16*pi*3,47 = 75,65 + 174,42 = 250,07 (0,07 of mistake due some rounding up)


Tips appreciated :D

Quote (Churika @ Jul 7 2023 04:30pm)

I know we are on jsp, but I'm really not a fan of just straight up giving the answers ; math is really more about the thought process and by giving them the answer straight up you're robbing them of that ! Really more of a fan of just asking questions until they figure it out (but like I said, I know I'm on jsp forum so I'm probably talking in the wind and not pointing at you in particular)

Quote (NomadJe @ Jul 7 2023 10:37am)

IRL I'm like you depict we should be, here the interaction is very limited and it's more a give-give on demand community.

Quote (Churika @ Jul 7 2023 04:39pm)

Yeah I completely understand ! I guess I'm just a bit sad so many people hate math related subject, when for them it's just a mean to end, when really math should be thought of as an art. Nothing against you / anyone in particular of course

Quote (NomadJe @ Jul 7 2023 11:37am)

Well, I wish you didn't have anything against a professional mathematician :thumbsup:

Quote (Churika @ Jul 11 2023 02:44am)

I'm curious ; what's a "professional mathematician"?