For our project, we used Mathematica to make a model of a taro milk tea with boba that you could get at a boba shop. The parts for the boba tea required a cup, tea, boba pearls, ice cubes, a top for the cup, and a straw. For the cup, we wanted to make it slanted as how most commercial cups are, so we needed to make part of a cone and a disc for the bottom of the cup. We decided to make a cup with a bottom diameter of 6 units, a top diameter of 10 units, and a height of 12.5 units. To create a cone, we had to use spherical coordinates. Therefore, we needed to find the rho, phi, and theta based on our desired measurements.
For the tea, we didn’t want a cup filled to the top. Instead, we decided to make the tea 8 units high out of 12.5 units of the cup. Another cone like the cup’s shape was made, so the tea would fit perfectly in the cup. For the boba pearls, spherical coordinates had to be used in order to make 21 in total. To make a sphere with a radius of 0.5 unit., we had rho = 0.5, phi from 0 to Pi inclusive, and theta from 0 to 2Pi inclusive. Since each sphere has a diameter of 1 unit. and we wanted a 0.05 unit space between each pearl (did not want smushed pearls), each sphere was moved by multiples of 1.05 units in some direction. For the ice cubes, we decided to make 4 of them. To make one that is 2 units by 2 units by 2 units, 6 planes in Cartesian coordinates had to be created. For each ice cube, there were 2 planes that had different x coordinates with a difference of 2 units. For the straw, we used cylindrical coordinates to make a long cylinder. We wanted the straw to be able to fit one of the boba pearls (one pearl has a radius of 0.5 unit), so we made the straw a tiny bit larger with a radius of .55 unit. For the top of the cup, we used polar coordinates to make a flat disc with a hole in the center. We designed the top to fit the straw and slightly hang over the sides of the cup.
