A 30.0-cm -long cylindrical plastic tube, sealed at one end, is filled with acetic acid. The mas

step 1 Here we have a cylindrical glass tube. We can see that a tight is 30 centimeters long, and it is

A 30.0-cm -long cylindrical plastic tube, sealed at one end,...

Key Concepts

Algebraic Manipulation

Solving for an unknown dimension such as the diameter involves isolating the variable in the formula derived from combining physical and geometric relationships. This involves rearranging equations and applying algebraic steps to solve for the desired quantity.

Cylindrical Geometry

The volume of a cylinder is calculated using the formula V = ?r²L, where r is the radius of the circular cross-section and L is the length of the cylinder. Understanding this geometric relationship is crucial when dealing with problems that require finding dimensions of cylindrical objects.

Volume Calculation

Determining the volume occupied by a substance using its mass and density is a key step in many problems. By rearranging the density formula (volume = mass/density), one can compute the volume of a liquid or solid, which is necessary for further geometric analyses.

Density

Density is the physical property that relates the mass of a substance to its volume. It is expressed as mass per unit volume and is essential for converting between the two. In problems involving density, the equation density = mass/volume is used to find any one of the three variables if the other two are known.

Transcript

00:01 Here we have a cylindrical glass tube.

00:03 We can see that a tight is 30 centimeters long, and it is going to be full of acetic acid, a formula c -h -3 -c -o -o -h.

00:14 And we want to know is the diameter of the actual tube.

00:21 And so another couple of things that we know is that in order for us to fill the whole tube with cetic acid, it's going to take, let's go ahead and type this up here.

00:30 It's going to take 89 .24 grams of acetic acid for the tube to be filled.

00:47 And then the density of acetic acids, we can put density here, is it's going to be 1 .05 grams per milliliter.

00:59 So knowing these, we can definitely try to find the diameter of the actual tube.

01:06 And it's going to take us a couple of steps.

01:08 What we're going to get there.

01:10 What we're first going to need to do is going to find the volume of the tube.

01:14 From the volume, we're going to be able to find the radius of the tube.

01:19 And from the radius, we can determine the diameter.

01:22 So we can start with the volume of the tube.

01:24 And we're going to use the 89 .24 grams of acetic acid to start.

01:30 So what we can do is do we're going to do a little bit of dimensional analysis.

01:35 We're going to do 89 .24.

01:39 Let me do that a little bit better, 89 .24 grams.

01:48 And we know that we're going to use the density to convert to milliliters.

01:53 We know that that's also going to allow us to cross out the grams.

01:56 So we have 89 .24 grams of acetic acid, and we have the density of acetic acid, which is 1 .2.

02:03 0 .05, we put a little dot here, grams per milliliter, and we can convert milliliters to cubic centimeters, and that's going to be a measurement of volume.

02:16 And we know that for every one, every milliliter, we have one cubic centimeter.

02:26 And when we multiply that out, so we're basically going to be dividing 89 .24 by 1 .05, and we get 84 .9 .9 .9 .5.

02:35 And we get 84 .9 nine centimeters.

02:44 Let me put a little decimal point there.

02:46 That's going to be our volume measurement, of the actual two.

02:51 And using this volume, we can go ahead and determine the radius.

02:56 And the way which we determine radius is by using the volume of a cylinder formula.

03:02 And the volume of a cylinder formula is this going to be, if you recall, v is equal to pi.

03:11 There's no pi symbol here, but i'm just going to put pi, r squared, times the height.

03:20 It's going to be the formula that we're going to use to find the radius.

03:23 So we're just going to be arranging a couple things because we want to find the radius here.

03:28 We have the volume.

03:30 We know what pi is, and we know the height, which is going to be 30 .0 centimeters.

03:35 So we're going to rearrange this so it can equal r squared.

03:39 So by arranging this formula just a little bit, in order for it to equal r squared, and then eventually convert it to r.

03:47 So we're going to do r squared here.

03:49 We're going to do this little step by step, r squared...

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