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Maths

the last question of my assignement (10 questions) which I have managed to complete 9 (with the help of this forum) - this one has stumped me

A can of orange juice is in the form of a right-circular cylinder and holds a volume of 1000 cm^3 of fluid. what dimensions of the can will give the smallest surface area?

i've worked this out:

Pi * r^2 * h = 1000
S.A Top = Pi * R^2
S.A Bottom = Pi * R^2
+ Cylinder Wall = 2 * Pi * r * h

Surface Area Total = Pi * R^2 + Pi * R^2 + 2 * Pi * R * H

Volume = Pi * r ^ 2 * h
Volume = 1000
Height = 1000 / (pi * r ^ 2)

SA(r) = 2*pi*r^2 + 2000/r

but dont know where to go from here - can anyone help? This is my last assignemnt for the year! Happy dayz

Louk
Reply 1
Avatar for Barny
Barny
17
Whatsa right circular cylinder?
Reply 2
Avatar for Louk
Louk
OP
no idea - just how the question is phrased

I assume it means a normal cylinder

Louk
Reply 3
Avatar for jyuk
jyuk
I had a go, and seem to have found a maximum surface area. But no minimum.
Reply 4
Avatar for Louk
Louk
OP
Cool - How did you get that??

Louk
Volume = (Pi)r^2(h) = 1000

h= 1000/(pi*r^2)



Surface Area = 2(pi)r^2 + 2(pi)rh

Substitute expression for H into formula:
S.A. = 2(pi)r^2 + 2(pi)r* (1000/(pi)r^2)

Then differentiate this formula (now only one unknown - r)and find a minimum turning point.
Use this value for r to calculate the dimensions.

Hope this helps!
I just did it very quickly, so may have made a mistake, but I got
r=5.42
h=10.84
Reply 7
Avatar for Ralfskini
Ralfskini
10
hitchhiker_13
I just did it very quickly, so may have made a mistake, but I got
r=5.42
h=10.84



Thats right!
Reply 8
Avatar for jyuk
jyuk
Louk
Cool - How did you get that??

Louk

Dunno, must have went wrong somewhere.:confused:
Ralfskini
Thats right!



Well yay for me!
:tongue:
Reply 10
Avatar for Jonatan
Jonatan
13
Louk
the last question of my assignement (10 questions) which I have managed to complete 9 (with the help of this forum) - this one has stumped me


Which level of Maths are you doing?

In general when you want to find maximum and minimum points of a function you first find the derivative of the function and then solve for 0. This point will then be a maximum or a minimum. That this is teh case can easily be illustrated by considering the motion of a stone throw vertically upwards into the air. At the moment the stone has reached its highest point it is (for an infinetely small amount of time) standing still. AFter that it starts moving down. The derivative pf the stones position with respect to time is zero when the stone has just reached its highest point.

Examples:

Find the maximum/minimum of the following functions:

F(x) = -x²
G(x) = x² + x + 4
H(x) = sin(x)

Solutions:

F'(x) = -2x = 0
Hence x = 0.

G'(x) = 2x + 1 = 0
Hence x = 1/2

H'(x) = cos(x) = 0
Hence x = 90 degrees = 0.5pi radians

I am not going to explain how to find the function for the surface area as a function of r as others have already done so, but once this function has been found this is the procedure to follow.
i dont understand how H(x) = sin x => H'(x) = cos x
actually im kinda lost on the whole equation from start to finish, H(x) :frown:~
Reply 12
Avatar for Nylex
Nylex
10
Revelation
i dont understand how H(x) = sin x => H'(x) = cos x
actually im kinda lost on the whole equation from start to finish, H(x) :frown:~


H'(x) is the derivative of H(x)...
Revelation
i dont understand how H(x) = sin x => H'(x) = cos x
actually im kinda lost on the whole equation from start to finish, H(x) :frown:~



its a long and boring proof. dont worry about it.

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