The shape of a custom-made gold wedding band is obtained by revolving the band about the vertical axis. The resulting wedding band has inner radius R, minimum thickness T, and width W. The outer boundary of the ring (the curved surface) is formed by the arc of a circle whose center lies on the axis of symmetry. For a standard wedding band, R might be anywhere from 6 to 12 mm, T might be 0.5 to 1.5 mm, and W might be 4 to 10 mm. If a customer asks the price of a wedding band with given dimensions R, T, and W, the jeweler must first calculate the volume of the desired band to determine how much gold will be required to make it.

Suppose that the jeweler charges the customer $1000 per troy ounce of alloy (90% gold, 10% silver) used to make the ring. (The profit on the sale, covering the jeweler's time and overhead in making the ring is fairly substantial because of the price of gold is generally under $400/oz and that of silver under $5/oz) The inner radius R of the wedding band is determined by measuring the customer's finger. The jeweler always makes wedding bands with T=1mm. Then, for a given acceptable cost C (in dollars), the customer wants to know the maximum width W of the wedding band he or she can afford.

The circumference of the finger is 60mm. The amount the consumer is willing to pay for the ring is $5,000. Using the equations V=(piW/6)(W^(2)+12RT+6T^(2)), with T=1, to find the width of the band that corresponds to the amount willing to pay. The density of the gold-silver alloy is 18.4 gm/cm^(3) and 1 lb contains 12 troy ounces and 453.59 gmn