Shire Silver Solutions

We need to be able to calculate the exact dimensions of the pieces we are going to make, in order to ensure proper mass. The dimensions we will need are:

• depth
• width of face
• height of face
• radius of corners

The density of silver is 10.49 grams/cm³, and we want to end up with 10 grams. We need to calculate the volume of silver that will make 10 grams. We divide 10 by 10.49 to get 0.95329 cm³.

We have decided to go with 22ga (gauge), which is .02534 inches, or 0.643636 millimeters or 0.0643636 centimeters. Dividing the previous result (0.95329 cm³) by the depth (0.0643636 cm) gives us the area of the face: 14.811 cm².

Now we need to calculate the dimensions of the face that will meet that area. To do this, we decompose the face into components:

The light blue area is a square. Let's call the length of the sides L. The pink areas are sized by their radius (R), and combining them gives us a full circle. The yellowish rectangles are sized by the light blue square and the radius of the rounding areas which gives us L*R. So adding these all up, we end up with L² + 4LR + πR².

Now we have a formula for the desired area with two variables: 14.811 = L² + 4LR + πR².

If we assume the dimension specified here, which is the outer square dimension, we can calculate the radius. First we need to reduce the outer square length to the inner square length by subtracting the radius. This gives us 14.811 = (3.89 - R)² + 4(3.89 - R)R + πR² ==> oops, this leads to a negative radius! We need to do this another way.
This gives us 14.811 = (3.89 - 2R)² + 4(3.89 - 2R)R + πR².

Let's define the radius as a fraction of the inner square's side length: R = xL. Then our formula becomes 14.811 = L² + 4LxL + π(xL)² ==> 14.811 = L² + 4xL² + πx²L².

Using 10% as the ratio, we get 14.811 = L² + 4(0.1)L² + π(0.1)²L² ==> 14.811 = L² + 0.4L² + 0.0314159L² ==> 14.811 = L² + 0.4L² + 0.0314159L² ==> 14.811 = 1.4314159L² ==> 10.347 = L² ==> L = 3.217 cm, and R = 0.3217 cm. This makes the outer square approximately 3.8 cm, which sounds about right.

We could try other values for x, and values between 0.05 and 0.15 seem reasonable; but I've had enough math for today.

So, the calculated dimensions are:

• depth: 0.0643636 centimeters (22 gauge)
• width of face: inner square=3.217 cm, outer square=3.8604 cm
• height of face: inner square=3.217 cm, outer square=3.8604 cm
• radius of corners: 0.3217 cm

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