Surface Area of Cylinders?

Question

given that the volume of a cylinder must be 1000 mL cubed...is there are any way to logically determine the dimensions of one that will reuslt in the lowest possible surface area outside of guess and check?
ie) Assuming a radius and then detrmining the resulting height and SA

If there is any theory or logic involved kindly let me know. Your assistance is appreciated

Answer 1

Need a little more info to formulate something, but remember that if it is a cylinder,

V = Pi (D) H or Pi (R2) H
R2 is r squared

where:
V=Volume
Pi = Pi :)
D = Diameter
R = Radius
H = Heighth or Length of the Cylinder So, your formula would start out like this

1000/PI = DH or (R2)H

Answer 2

1) Assume cylinder means an object of constant circular cross-section and length whose centerline is always normal to its cross-sectional area.

2) The surface area of a cylinder is a minimum when its radius is a maximum

3) Proof of 2)

3a) The surface area of any cylinder, A, equals 2 * L * PI * R; where L is the length and R is the radius.

3b) The volume of any cylinder, V, equals L * PI * R^2

3c) Eliminating L from A yields A = 2 * PI * R * (V / ( PI R^2)) = 2 * V / R;

3d) So, by 3c), the surface area of a constant-volume cylinder is inversely proportional to the radius - this proves 2).

Answer 3

V = 1000 = pi*r^2*h where r is radius and h is height of cylinder.
Therefore h = 1000/(pi*r^2) <-- Eq 1
Surface area = S = 2*pi*r *h+2*pi*r^2 <-- Eq 2
Substitute 1000/(pi*r^2) from Eq 1 for h in Eq 2, getting:
S = 2*pi*r(1000/(pi*r^2)) +2*pi*r^2
S = 2000/r + 2*pi*r^2
dS/dr -2000/r^2 +4pi r
Set derivative to 0
-2000/r^2 +4pi r =0
-2000 +4pi r^3 = 0
r^3 = 500/pi
r = 5.4193 radius for minium surface area
If you figure it out this will be exactly 1/2 the height or the diameter = height.

Answer 4

The one with the Diameter equal to the height gives the smallest surface area for the volume. Im pretty sure chek and see though.

Answer 5

im no longer gonna provide the reply : P you want to study ! heres help even with the actuality that :) to locate the exterior are of the finished cylinder you want to operate the exterior portion of both circles and the exterior portion of the different (rectangular-ish) structure. circle floor section = pi (3.14) situations the radius (2.5) situations the radius (2.5) multiply that answer with assistance from 2 so that you get the realm of the proper AND bottom circles different floor section = circle circumfrence [diameter (radius situations 2 .. 18.6 + 18.6) situations pi (3.14) upload that determination to portion of the circles and theres your answer !! desire that helped !! :)

Answer 6

V = (pi)(r^2)(h)
SA = (2r)(pi)(h) + 2(pi)(r^2)

1000 = (pi)(r^2)(h)
h = 1000/(pi*r^2)

Put that into the SA equation:
SA = 2r(pi)(1000/pi*r^2) + 2(pi)(r^2)
= 2000/r + 2(pi)(r^2)

Now that you have an equation in two variables, you can minimize or maximize it as usual. (At the moment, I can't remember how to min/max a function...)

Answer 7

Go to a referance page Duh lol
you should print it out to