Problem solving

Question

Each side length of a unit cube measures 2 units and increases by 50% every minute. What is the ratio of the surface area after 3 minutes to the original surface area?

Answer 1

The ratio of the surface area after 3 minutes to the original surface area is 729/64.

The formula for the surface area of a cube is A = 6s². Initially, each cube's side is 2 units, so if we plug that into the formula, the initial surface area of the cube is A_initial = 6(2)² = 24 units². Since we know that each side length of the cube increases by 50% each minute, we can say that s_new = s_initial(1 + r)^t, where r is the ratio each side increases by, t is the number of minutes from the cube's initial state, s_initial is the initial length of the cube's side, and s_new is the new length of the side. If we want to know the new side length of the cube after 3 minutes, we can plug in these values into the formula so that s_new = 2(1 + 0.5)³ = 6.75 units. Then, to find the new surface area of the cube, we can plug this new s value into the formula so that A_new = 6 (6.75)² = 273.375 units².

Since we want the ratio of the new surface area to the old one, we can simply say that A_new/A_initial = 273.375/24. Since we cannot have decimal values in a ratio, we can multiple both the numerator and the denominator by (8/3) so that the ratio becomes (273.375)(8/3) / (24)(8/3) = 729/64. Therefore, the ratio of of the surface area after 3 minutes to the original surface area A_new/A_initial = 729/64.