# Equation: Surface area is numerically equal to volume

• Apr 21st 2011, 01:13 PM
Lewis1
Equation: Surface area is numerically equal to volume
The problem is as follows:
The total surface area of a cylinder is numerically the same as its volume.
The radius of the cylinder is rcm, the height is hcm.
Express h in terms of r.

The answer is h = 2r / (r – 2)

Surface area of cylinder = hπ2r (π is the symbol for pi)
Volume of cylinder = hπr^2

If the volume and area are numerically the same then I assume

hπ2r = hπr^2

By dividing both sides by π

h2r = hr^2

At this point I am stuck (assuming the previous steps are correct). Can anyone help?

Lewis
• Apr 21st 2011, 01:57 PM
Quacky
Surface Area of a Cylinder isn't that. It's the area of the two circular sections (2πr^2) Added to the longitudinal area (I don't know what this is called) which is 2πrh

2πr^2+2πrh = πr^2h

And solve for h.
• Apr 21st 2011, 02:34 PM
Lewis1
Thanks Quacky and Topquarks,
Yes I did forget the end caps. I assumed it was a hollow cylinder.

Here is the solution if anyone is interested

2πr^2 + 2πrh = hπr^2

Divide by π

2r^2 + 2rh = hr^2

Bring all the h terms onto one side

2r^2 = hr^2 - 2rh

Isolate the h term

2r^2 = h(r^2 - 2r)

2r^2 / (r^2 - 2r) = h

Simplify the form of the denominator

2r^2 / r(r - 2) = h

Cancel (divide) the r’s in the numerator and denominator

2r / (r - 2) = h

• Apr 21st 2011, 04:58 PM
Wilmer
Quote:

Originally Posted by Lewis1
2πr^2 + 2πrh = hπr^2
Divide by π
2r^2 + 2rh = hr^2

Quicker if you divide by r:
2r + 2h = hr
hr - 2h = 2r
h(r - 2) = 2r
h = 2r / (r - 2)
• Apr 22nd 2011, 04:20 PM
HallsofIvy
Quote:

Originally Posted by Lewis1
The problem is as follows:
The total surface area of a cylinder is numerically the same as its volume.
The radius of the cylinder is rcm, the height is hcm.
Express h in terms of r.

The answer is h = 2r / (r – 2)

Surface area of cylinder = hπ2r (π is the symbol for pi)
Volume of cylinder = hπr^2

If the volume and area are numerically the same then I assume

hπ2r = hπr^2

By dividing both sides by π

h2r = hr^2

At this point I am stuck (assuming the previous steps are correct). Can anyone help?

If this were a cylinder without ends, so that this equation was correct, divide both sides by hr.

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