Question

1. suppose you have 2 pieces of ice with the same volume but in different shapes. if one of the pieces has a greater surface area than the other, it will cool a beverage faster than the ice with less surface area.

a. describe 2 different pieces of ice that have the same volume, but have different surface areas.

b. which piece of ice will cool a beverage faster?

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Explanation:

• For part a: Consider geometric shapes. A sphere and a cube can have the same volume. The formula for the volume of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\), and for a cube \(V_{cube}=s^{3}\). If we set \(\frac{4}{3}\pi r^{3}=s^{3}\), we can find corresponding values of \(r\) (radius of sphere) and \(s\) (side - length of cube). The surface area of a sphere is \(A_{sphere} = 4\pi r^{2}\), and the surface area of a cube is \(A_{cube}=6s^{2}\). Since the relationship between \(r\) and \(s\) from the volume equality does not lead to \(4\pi r^{2}=6s^{2}\), they have different surface areas. Another example could be a rectangular prism with dimensions \(l\times w\times h\) and a cylinder with radius \(r\) and height \(h_{cylinder}\). If \(l\times w\times h=\pi r^{2}h_{cylinder}\) (volume equality), their surface areas \(A_{prism}=2(lw + lh+wh)\) and \(A_{cylinder}=2\pi r(r + h_{cylinder})\) will generally be different.
• For part b: Heat transfer (which causes the cooling of the beverage) occurs at the surface of the ice. A larger surface area means more contact between the ice and the beverage. According to the principle of heat transfer (Newton's law of cooling in a simplified sense for this context), more surface area allows for more heat to be absorbed from the beverage in a given time.

Answer:

a. One could be a spherical - shaped ice piece and the other a cubic - shaped ice piece. (Or other valid geometric shape combinations like a cylindrical - shaped ice and a rectangular - prism - shaped ice as long as volume is equal but surface area is different).
b. The ice piece with the greater surface area.