How Do You Calculate the Radius of a Cylinder from Its Label Area and Height?

[neuro.akn]

Homework Statement

The area of a label on a cylindrical can is defined by

A = 16(pi)x^2 - 68(pi)x + 42(pi). If the height of the can is defined as

h = 2x - 7, determine the radius of the can.

*Note: the (pi) represents a pi symbol; it would not work on my computer*

Homework Equations

 

[Jufro]

Let's imagine taking the label off of the can and unrolling it.

Then we should end up with a rectangle whose area = L*W.

But the width= height of the label= 2x-7

and the length= the circumference= 2 (pi) r.

Make sense now?

 

[neuro.akn]

Thank you! This is starting to make sense now, but what would I then need to do with the equation given for the area of the can?

 

[neuro.akn]

I'm not sure how I would solve for the radius, even though what you said makes sense.

 

[Jufro]

If we set the two areas equal to each other:

(2x-7) * 2(pi)r = 16(pi)x^2 - 68(pi)x + 42(pi)

Solving for r leads would mean that we have to divide both sides by (2x-7) * 2(pi).

The long division you can try yourself but that will leave you with the radius on the RHS.

 

[haruspex]

I'm not sure how I would solve for the radius, even though what you said makes sense.

You have two expressions for the area, one given involving only x, and now another involving r and x. Equate them.
The answer will be in the form r = some function of x. Don't expect to get a number for r.

 

[neuro.akn]

Here is how I am solving:

(2x-7)*2(pi)r = 16(pi)x^2 - 68(pi)x + 42(pi)

I divide each side by (2x-7)*2(pi) but this expression (2x-7)*2(pi) simplifies into -10x.

So I will proceed to long divide 16(pi)x^2 - 68(pi)x + 42(pi) by -10x ?

 

[haruspex]

Here is how I am solving:

(2x-7)*2(pi)r = 16(pi)x^2 - 68(pi)x + 42(pi)

Yes

this expression (2x-7)*2(pi) simplifies into -10x.

it does?! How?

So I will proceed to long divide 16(pi)x^2 - 68(pi)x + 42(pi) by -10x ?

No, long divide by (2x-7)*2(pi).

 

[neuro.akn]

When you multiply (2x-7)*2(pi), you get -10x.

Consider:
(2x-7)*2(pi)
= 4(pi)x - 14(pi)
= -10x

 

[BobG]

Try it one term at a time. Pi obviously divides out of both sides, etc.

After a couple of steps, one side of the equation is going to be a polynomial that you have to factor.

Long division works.

So does FOIL if you have reason to believe the problem probably isn't particularly difficult and that one of the terms of the polynomial will probably be equal to your height. There's only so many possibilities that you can plug into the other term just using a trial and error method. (In fact, it's not a particularly difficult polynomial to factor even if you didn't already know one of the terms.)

 

[BobG]

When you multiply (2x-7)*2(pi), you get -10x.

Consider:
(2x-7)*2(pi)
= 4(pi)x - 14(pi)
= -10x

No offense, but this is really, really wrong and I take back my statement that you'll wind up with a polynomial that's easy to factor.

You can't subract 14(pi) from 4x(pi) and get -10x for a couple of reasons.

If your equation was 4(pi) - 14(pi), your answer would be -10(pi); not -10. Subtraction doesn't cancel out pi.

You can't subtract 14 from 4x and get -10x unless x happens to be 1 or 0 and even then, it would only be coincidence that your answer was correct because you performed an illegal operation.

In fact, if you're at all in doubt that this doesn't work, just let x=2 and do the subtraction you just did. (Whew! Right after I posted this, I looked at the equation and thought, "NO WAY!" Fortunately, if you let x=2, you get -18.85; not -20.)

You could subtract 14x from 4x or you could subtract 14 from 4, or you could subtract 14x^2 from 4x^2, etc.