SOLUTION: A soup can that is a cylinder has a radius of 2x - 1 and a height of 3x. What is the surface area of the soup can?

Algebra ->  Rational-functions -> SOLUTION: A soup can that is a cylinder has a radius of 2x - 1 and a height of 3x. What is the surface area of the soup can?      Log On


   

Question 596005: A soup can that is a cylinder has a radius of 2x - 1 and a height of 3x. What is the surface area of the soup can?
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--
.
The surface area of a soup can (a cylinder) includes three areas: the top of the can (a circle), the bottom of the can (also a circle), and the side of the can (a rectangle.) We will calculate the areas of each of these and add them together to get the total surface area. Our answer will be an expression in terms of x.
.
.
[I] The surface area of the top and bottom of the can
.
Notice that the top and bottom of the can have the same area. The circle has a radius of 2x-1. We use the formula for the area of a circle, and substitute 2x-1 for r. We'll let A be the area, with subscripts t and b for the top and bottom.
A%5Bt%5D=%28pi%29%28r%5E2%29
A%5Bt%5D=%28pi%29%282x-1%29%5E2
.
A%5Bb%5D=A%5Bt%5D=%28pi%29%282x-1%29%5E2
.
.
[II] The surface area of the side of the can
.
If it was laid out flat, the side of the can would be a rectangle. The dimensions of the rectangle are the height of the can and the circumference of the circle. To get the area, we multiply these dimensions together. We'll use the formula for the circumference of the circle times the height h. We'll use the subscript s for this area.
A%5Bs%5D=%282%29%28pi%29%28r%29%28h%29
A%5Bs%5D=%282%29%28pi%29%282x-1%29%283x%29
.
.
[III] The total surface area
.
We'll together the three areas to get the total surface area.
A=A%5Bt%5D%2BA%5Bb%5D%2BA%5Bs%5D
A=%28pi%29%282x-1%29%5E2%2B%28pi%29%282x-1%29%5E2%2B%282%29%28pi%29%282x-1%29%283x%29
.
Now we will do some nifty algebra on this expression to simplify it. First combine the area of the top and bottom of the can.
A=%282%29%28pi%29%282x-1%29%5E2%2B%282%29%28pi%29%282x-1%29%283x%29
.
Now, if you look carefully, you'll see that both the remaining terms have %282%29%28pi%29%282x-1%29 in them. We will factor that out and multiply it by the remaining factor.
A=%28%282%29%28pi%29%282x-1%29%29%28%282x-1%29%2B3x%29
.
Simplify by combining 2x and 3x.
A=%28%282%29%28pi%29%282x-1%29%29%285x-1%29
.
Personally, I would probably stop here, but you can also multiply these factors out to give a polynomial. You will know best what sort of answer your teacher is looking for.
.
A=%282%29%28pi%29%2810x%5E2-5x-2x%2B1%29
A=%282%29%28pi%29%2810x%5E2-7x%2B1%29
A=20%28pi%29x%5E2-14%28pi%29x%2B2%28pi%29
.
I hope this helps! Feel free to email if any part of the explanation doesn't make sense.
.
Ms.Figgy
math.in.the.vortex@gmail.com