wedding.html

Shopping for a Wedding Ring*

Department of Mathematics

University of Georgia

Athens, Georgia

Carol W. Penney

You are shopping for a wedding band, which you will have made by a famous goldsmith. He offers these four styles of rings in his catalog:

round cross-section,
D cross-section,
court section (sometimes known as "comfort fit"), with a cross-section in the shape of an ellipse, and
the unusual but comfortable reverse D section.

Your jeweler deals internationally, so he has to deal with various ring size scales of measurement, including the British, American, Japanese, and European sizes. American sizes are based upon the circumference of the inside of the ring, with U.S. size 0 having a circumference of 36.3 mm and each size increase equivalent to 2.60 mm of circumference. If you know your U.S. ring size , you can calculate the circumference of your finger as follows: . Otherwise measure your ring finger by wrapping a string around it and measuring the length of the string. The circumference of your ring finger probably lies in the range of 45 to 70 mm.

The length of a ring is the length of finger covered by the ring. Round rings are not as comfortable between your fingers as are the other types, so if you buy a round cross-section ring, you will buy one that is fairly short, say 3 mm long. If you buy either a D-shaped ring or a court ring, you want it to be between 3 and 6 mm long. The catalog gives the cost of only one ring: the circular cross-section ring with length 3 mm in size 6. It costs $229.

Determine the volume of the round cross-section 3 mm ring, and use this volume to determine the cost per cubic millimeter of this goldsmith's rings.
Select your favorite of the three other styles (not the round style) , choose the length of the ring that you prefer in each style, then calculate the volume and price of this ring in your size and desired length.

Note: Due to a round-off error, you may have difficulty integrating from the center of a circle to an edge. For example, if Maple approximates using an approximation of x=sqrt(2), it may use a value of x slightly larger than sqrt(2), resulting in a negative number under the square root. If it then integrates, it may get complex numbers for an answer. You can avoid this problem by integrating slightly to the left of x=sqrt(2); say from x=0 to x=sqrt(2)-.0001, instead of from x=0 to x=sqrt(2). Your integral will be approximately correct and will not then involve complex numbers.

Here are diagrams and specifications for each of the four styles:

Round Cross-Section 3 mm ring in size 6:

The radius of your finger is the radius of the inside of the ring. The radius of a size 6 ring is approximately 8.25 mm, as shown by the following calculations.

> restart:with(plots):with(plottools):

> C:=36.3+2.6*6;R:=evalf(C/(2*Pi));

This ring is the solid generated by revolving a disk of radius 1.5 mm around a vertical line, the y-axis. This disk is bounded by the circle with equation . for . The simplest way to plot this circle is by using the implicitplot command, in which you state the equation and list the domain and range to be graphed. The view option widens the window, so that we can see the -axis about which we revolve the circle.

> L:=3;

> implicitplot({x^2+y^2=(L/2)^2,x=-R-L/2},x=-R-L/2..L/2,y=-L/2..L/2,
scaling=constrained,view=[-R-L/2-.5..L/2,-L/2..L/2],color=black);

Here is the ring:

> golden:=COLOR(RGB,.97,.73,.35):

> display(torus([0,0,0],1.5,R+1.5),scaling=constrained,color=golden,
lightmodel=light4,style=patchnogrid);

>

D-section:

This ring is described as the solid generated by revolving the right half of a disk about a vertical line. The following commands will show a lamina being revolved and a typical D-section ring.

> restart:with(plots):with(plottools):

> R := 8.260141544:L:=6:

> p1:=implicitplot({x=R,(x-(R))^2+y^2=3^2},x=R..R+3,y=-L/2..L/2,
scaling=constrained,view=[-.25..R+L/2,-L/2..L/2],thickness=2,
color=black,axes=none):

> p2:=implicitplot(x=0,x=0...1,y=-L/2..L/2,
scaling=constrained,view=[-.25..R+L/2,-L/2..L/2],thickness=2,
color=black,axes=none):

> display({p1,p2});

>

Here is a typical D-section ring:

> golden:=COLOR(RGB,.97,.73,.35):

> outside:=implicitplot3d((r-R)^2+z^2=3^2,r=R..R+L/2,t=0..2*Pi,
z=-L/2..L/2,coords=cylindrical,scaling=constrained,color=golden,
grid=[3,20,5],lightmodel=light4,style=patchnogrid):

>    inside:=plot3d(R,t=0..2*Pi,z=-L/2..L/2,coords=cylindrical,
  scaling=constrained,color=golden,grid=[30,2],lightmodel=light4,
  style=patchnogrid):

> display([inside,outside]);

>

Court section:

This ring is described as the solid of revolution obtained by revolving an ellipse twice as high as wide about a vertical line.

This ellipse will have equation .

> restart:with(plots):with(plottools):
R := 8.260141544:L:=6:

> a := 4/L: b := 2/L:

>    pic1:=implicitplot(a^2*(x-(R+L/4))^2+b^2*y^2=1,x=R..R+L/2,
  y=-L/2..L/2,scaling=constrained,view=[0..R+L/2,-L/2..L/2],axes=none,
color=black,thickness=2):
pic2:=implicitplot(x=0,x=0..R+L/2,
  y=-L/2..L/2,scaling=constrained,view=[0..R+L/2,-L/2..L/2],axes=none,
color=black,thickness=2):
display({pic1,pic2});

> golden:=COLOR(RGB,.97,.83,.42):

> implicitplot3d(a^2*(r-(R+L/4))^2+b^2*z^2=1,r=R..R+L/2,t=0..2*Pi,
z=-L/2..L/2,coords=cylindrical,scaling=constrained,color=golden,
grid=[20,20,20],lightmodel=light4,style=patchnogrid);

>

Reverse D-Section:

This ring is described as the solid generated by revolving the left half of a disk about the -axis. Here is a typical reverse D-section ring:

> restart:with(plots):with(plottools):

> R := 8.260141544:L:=6:

> pic3:=implicitplot({x=R+L/2,(x-(R+L/2))^2+y^2=3^2},
x=R..R+L/2,y=-L/2..L/2,scaling=constrained,view=[0..R+L/2,-L/2..L/2],
axes=none,color=black,thickness=2):
pic4:=implicitplot(x=0,
x=0..R+L/2,y=-L/2..L/2,scaling=constrained,view=[0..R+L/2,-L/2..L/2],
axes=none,color=black,thickness=2):
display(pic3,pic4);

>

Here is the ring:

> golden:=COLOR(RGB,.97,.77,.32):

> inside:=implicitplot3d((r-(R+L/2))^2+z^2=3^2,r=R..R+L/2,t=0..2*Pi,
z=-L/2..L/2,coords=cylindrical,scaling=constrained,color=golden,
grid=[3,15,5],lightmodel=light4,style=patchnogrid):

>    outside:=plot3d(R+L/2,t=0..2*Pi,z=-L/2..L/2,coords=cylindrical,
  scaling=constrained,color=golden,grid=[20,2],lightmodel=light4,
  style=patchnogrid):

> display([inside,outside]);

>

Discount:

You discuss your choice with the goldsmith. He is amazed that you predicted the cost so accurately until you tell him that you are a calculus expert. He has been trying to solve the following problem and excitedly offers you a 50 percent discount if you can solve it for him:

He makes beaded necklaces out of 18 K gold beads that he strings on a gold chain. He forms each bead by drilling a core through the center of a solid gold sphere. He keeps a collection of spheres of different sizes; he has 10, 10.5, 11, 11.5, and 12 mm beads from which the customer can select. He wants to use exactly 71 beads on each necklace, and he wants each necklace to have the same length, a total length of 702.9 mm, so he wants each drilled bead to occupy 9.9 mm of the chain. He needs to know two things: Depending upon the size of the bead that the customer chooses, what radius hole should he drill to result in a bead that is 9.9 mm long? What volume of gold beads will be in the resulting necklace? In particular, he would like to post a chart on his wall so that after the customer chooses the bead size he can look at the chart to determine the radius of the hole he will drill and the volume of gold in each bead and in the necklace. Of course, for the 50% discount you are happy to solve this problem and construct a spreadsheet for him to post in his studio.

For possible diameters of 10, 10.5, 11, 11.5, and 12 mm , calculate the radius of the core that he will drill to make each bead.
Calculate the amount of gold in each drilled bead and the amount of gold in the 71 beads in the necklace.
Make a spreadsheet that he can post in his studio.

Each drilled bead is formed by drilling a hole of radius from a sphere of diameter mm. Here is the lamina that generates such a bead:

Making your Spreadsheet:

To create a spreadsheet showing the radius to be drilled in each bead, the volume of each bead, and the volume of all 71 beads in the necklace, follow the following steps. In this discussion, M/2 represents the radius of the gold sphere, r represents the radius of the drill and v(r) represents the volume of each drilled bead, in cubic mm.

1. First, work the problem of finding the radius r of the drill in terms of M. Define the function v(r) giving the volume of each bead as a function of r. This volume may be expressed in the form of an integral. Compute this volume as a floating point (decimal) number.
2. Select Spreadsheet from the Insert menu. Note that the entry in the first row first column (A1) is highlighted.
3. Type Bead Size (notice that this appears at the top of the screen in what is called the edit field of the context bar ) and press the Enter key.
4. Click on the box in the first row, second column, B1. Type Drill Radius and press Enter . You now have headings for two columns of your table. Make headings for other columns similarly.
5. You want 5 rows, one for each bead size. Click just outside of the spreadsheet (about 0.01 inch) to get a drag box around it. Grab on one of the small black handles at the bottom of the box and drag it up or down to resize the spreadsheet.
6. Click on the box in column A, row 2 to highlight it. Type the size of the smallest bead, 10, and press Enter . Enter the other bead sizes in the rest of the first column.
7. Go to the top of second column and click on the box B2. Type r(~A2) and press enter. This is the command to evaluate the radius of the drill, for a bead of size given in the first column to the left of B2. Highlight the second column as follows: click, while holding down the Shift key, on that box B2 and on the last box in that column that you want to fill, beside the last filled box in the first column, or drag your mouse down that column. Select fill , down from Spreadsheet Menu .
8. Then fill the other columns. For example, for volumes, type v(~A2/2) in the third column, second row.
9. You can play around with various Spreadsheet menu items to size your spreadsheet, display or remove a border, and so forth, to make it look good.

In Summary: This project has two parts: first, calculate the volume of gold and price of the ring of your choice in your size, and second, earn your discount by creating a spreadsheet the jeweler can use to create and sell his beaded necklaces. Be sure to state your problems, show your work, and state your answers clearly.

The Most Common Maple Commands

* Information in this project was obtained from the following web site: www.amulet.co.uk/

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.

I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.

I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

Signed: