According to the Official Rules of Baseball, a baseball must...

According to the Official Rules of Baseball, a baseball must have a circumference not more than $9.25$ in or less than $9.00$ in and a mass not more than $5.250 \mathrm{z}$ or less than $5.00 \mathrm{oz}$. What range of densities can a baseball be expected to have? Express this range as a single number with an accompanying uncertainty limit.

Key Concepts

Density

Density is a fundamental physical property defined as mass divided by volume. It describes how much matter is contained in a given volume and is central to understanding material properties in physics and engineering.

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)?r³, where r is the radius. This formula is crucial when converting linear measurements, such as circumferences or diameters, into volumes, especially when dealing with objects that are approximately spherical.

Uncertainty Propagation

Uncertainty propagation is a method used to determine how measurement errors in individual quantities affect the final computed result. When several measurements with uncertainties (derived from rules or experimental limits) are combined in calculations like density, proper error propagation helps in quantifying the overall uncertainty in the outcome.

Transcript

0:00 All right.

00:01 So this problem is having us do some density calculations.

00:03 So we're given the circumferences and the masses of, we're given the circumference range and the mass range of baseballs that are, according to the official rule of baseball, the baseballs have to be that circumference within the range of that circumference and within the range of those masses.

00:24 So they want us to get a density range and figure out what the uncertainty is in the density range, right? so the range is going to have to go from the least dense ball to the most dense ball.

00:44 Right.

00:44 So let's remind ourselves that density is equal to mass over volume, right? so the most dense ball is going to have, so in order to have the highest density, you want to have the highest density, you want to have the high.

00:56 Highest mass and the smallest volume.

00:59 So that'll make the value of d the largest.

01:03 And if you want the lowest density, so you want the lowest density, you want the mass to be the lowest, and you want the volume to be the largest.

01:13 So the large volume in the denominator will make d as small as possible.

01:19 Right.

01:20 So that's all fine and dandy.

01:21 We're going to say that again in a few minutes.

01:23 But so we have the we have the circumferences and inches, and we have the masses and ounces.

01:31 So we have to turn the circumferences into volumes so we can calculate our densities, right? so let's do that first.

01:39 So assuming the baseball is a perfect sphere, right, the circumference of a sphere is 2 pi r, and the volume of the sphere is 4 thirds pi r cubed.

01:54 So for the sphere that has the circumference of 9.

01:58 Inches, right? so the radius is going to be equal to the circumference over 2 pi, right? so then if we plug in our density, sorry, if we plug in our circumferences, so the radius is going to be 9 .00 inches divided by 2 pi.

02:25 And this is going to be equal to the first radius.

02:28 So i'll call it r1.

02:30 So r1 is going to be.

02:32 Be 1 .432 inches.

02:45 We'll do the same thing for the second one, right? so 9 .25 inches over 2 pi.

02:55 I'll call this r2 and that radius is going to be 1 .472 inches.

03:07 So now let's calculate the volumes with these, right? so i'll do this on a separate page.

03:13 So the volume for one is going to be four -thirds pi 1 .432 inches cubed, and that's going to equal 12 .310 inches cubed.

03:40 The second volume, there's going to be four -thirds same thing right pie that radius is going to be 1 .472 inches cubed and that volume is going to give us 13 .365 inches cubed okay so let's go back to the first page right so the most dense ball most dense is going to have the greatest mass which is 5 .25 ounces and the smallest density.

04:22 So let me, sorry, and the smallest volume...

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