Constructing a Cylindrical Tube A rectangular piece of cardboard, whose area is 216 square centi

Constructing a Cylindrical Tube A rectangular piece of...

Key Concepts

System of Equations Derived from Geometric Constraints

Problems that specify multiple constraints, such as a fixed area for a rectangle and a fixed volume for the resulting cylinder, require setting up and solving a system of equations. This system encapsulates the relationships between the rectangle’s dimensions and the cylinder’s properties, enabling the determination of unknown quantities.

Transformation of a 2D Shape into a 3D Object

The process of converting a flat, two-dimensional shape into a three-dimensional object involves mapping the dimensions from the 2D figure onto the features of the 3D object. In this context, understanding how the area of a rectangle is distributed among the cylinder’s dimensions (the circumference and the height) is essential for solving problems that involve constructing a cylinder from a given sheet.

Lateral Surface Area and the Cylindrical Net

When constructing a cylinder from a rectangular piece of material, one of the rectangle’s dimensions becomes the height of the cylinder, and the other dimension becomes the circumference of its base. This idea of the cylindrical net, where a two?dimensional rectangle is ‘curved’ to form the curved surface of a cylinder, is fundamental for establishing the relationship between the original rectangle and the resulting cylinder.

Cylinder Volume Formula

The formula for the volume of a cylinder, V = ?r²h, is a key concept in understanding how the dimensions of a cylinder (its radius and height) are related to its capacity. This relationship is crucial when a problem provides the volume and requires one to derive or relate other dimensional properties of the cylinder.

Transcript

00:01 All right, so this problem is kind of tricky because there's more involved in what we're going to be doing.

00:06 There's actually three variables in the way that i'm setting it up.

00:09 So we have a rectangle.

00:11 I'm going to call the length x and y.

00:14 And so these length x and y multiply to 216 square centimeters.

00:23 Then we were told that one of the edges is wrapped around to, touch the other edge and create a cylinder.

00:34 Kind of like if you were to unwrap the wrapping on a can of food or soup or whatever, if you cut it and then unravel it, it would create a rectangle.

00:46 So this is that rectangle that wraps around the cylinder.

00:51 Now, the volume of the cylinder is given to us, 224 square, sorry, cubic centimeters.

00:59 And you find that by pi times the radius squared multiplied by the height.

01:05 Our height is y, in this case.

01:08 Our radius has the relationship with x that x is basically the circumference.

01:15 This right here is the circumference of this circle.

01:18 So if i multiply 2 times pi times this radius, that's equal to x.

01:24 So x over 2 pi is equal to 5.

01:30 And so then i'm going to substitute this in for this expression here and in the place of y i'm going to substitute an expression for y.

01:45 Y is the same thing as 216 divided by x so then in this equation we have pi times x over 2 pi squared multiplied by 216 x is equal to 224.

02:08 So this is the relationship that we're going to use.

02:13 So then on this side, i have pi.

02:18 This is x squared over 2 pi times 2 pi.

02:29 And then here times 216 over x equals 224.

02:37 All right, we've got a lot of different pieces moving around.

02:40 So let's figure out what to do next.

02:41 Next, i see a pi and a pi.

02:44 Let's cancel those out.

02:45 That's one hole.

02:47 We have x and x, assuming that x doesn't equal zero, which it can't be if we have area and volume, then we can get rid of the square and get rid of the x here.

02:56 We have four here, so let's divide this by four.

03:00 That's 54.

03:01 All right, so we've got very little left.

03:06 That's good.

03:08 54 times x over pi is equal to 224.

03:15 Then we can multiply by the reciprocal here, pi over 54.

03:27 So this is x equal to.

03:31 I know that 2 makes 54, so i could divide this by 2 and this by 2.

03:36 It's 112, 27.

03:42 I think that's it because this is not divisible by 3, and this is divisible by a lot of 3s.

03:48 So 112 pi over 27 centimeters.

03:54 So this is the value for x that will make this work.