How do I find the radius of an aluminium can that has a height of 12 inches and a surface area of 400 sq.in.?

Answer 1

400 = 2 π r ² + 2 π r h
400 = 2 π r ² + 24 π r
(400 / 2π) = r ² + 12 r
64 = r ² + 12 r
r ² + 12 r - 64 = 0
(r + 16) (r - 4) = 0
r = 4
Radius = 4 inches (to nearest whole number)

Answer 2

Plug in the height and surface are to this formula and work it backwards to get the radius.
Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h


(h is the height of the cylinder, r is the radius of the top)

Surface Area = Areas of top and bottom +Area of the side

Surface Area = 2(Area of top) + (perimeter of top)* height

Surface Area = 2(pi r 2) + (2 pi r)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.

You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r2). Since there is both a top and a bottom, that gets multiplied by two.

The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.

Add those two parts together and you have the formula for the surface area of a cylinder.

Surface Area = 2(pi r 2) + (2 pi r)* h

Answer 3

If the surface area includes the two round caps(ends) then use the equation:-
Surface = 2A + Ch
Where 'A' is the area of the can ends and 'C' is the circumference, and 'h' the height.
A = pi r^2 (area of a circle)
C = 2 pi r (circumference of a circle)
So substituting in
Surface = 2 pi r^2 + 2 pi r h.
400 = 2 pi r^2 + 2 pi r 12
Now factorise
400 = 2pi(r^2 + 12r)
Divide through by 2pi
400/2pi = r^2 + 12r
r^2 + 12r - 400/2pi = 0
This is a quadratic equation and use 'completing the square to find 'r'.
r^2 + 12r - 63.66197724 = 0
r = -12 +/- sqrt[12^2 - 4(1)(-63.66)[/2(1)
r = -12 +/- sqrt[144 + 254.64]/2
r = -12 +/- sqrt398.64]/2
r = (-12 +/- 19.97)/2
r = (-12 +19.92)/2 = 3.98 (correct answer)
r = (-12-19.92)/2 = -15.96 (impossible answer - radii are always positive).

Answer 4

The height of 12 inches has nothing to do with the radius unless you are trying to calculate the volume of the can in cubic inches. You have an area of 400 sq. in. so you take the square root of the Area divided by Pi (3.14) r= sqrt A/3.14 and that gives you the radius. If you want the volume just multiply the area(400 sq. in.) times the height (12 in) and you get 4, 800 cu. in. That is a big aluminum can!

Answer 5

400 in² = π x D/2 x 12"
400² = 3.14 x D/2 x 12
400² = 37.7D/2
D/2 = √400/37.7 = √10.62/2" = 2.3"radius