Question

An ice cube is freezing in such a way that the side length s, in inches, is s(0) 1)=2+2, +2, where t is in hours. The surface area of the ice cube is the function A(s) = 6s.

Part A: Write an equation that gives the volume at t hours after freezing begins.

Part B: Find the surface area as a function of time, using composition, and determine its range.

Part C: After how many hours will the surface area equal 216 square inches? Show all necessary calculations.

Answer 1

The volume of the ice cube at t hours after freezing begins is given by V(t) = (2+t)^3. The surface area as a function of time is A(t) = 12+6t. The surface area will equal 216 square inches after 34 hours.

Step-by-step explanation:

Part A: To find the volume at t hours after freezing begins, we need to use the formula V(s) = s^3, where s is the side length of the ice cube. Given that the side length is s(t) = 2+t, we can substitute this expression into the formula to get V(t) = (2+t)^3.

Part B: To find the surface area as a function of time, we need to use the composition of functions. The surface area of the ice cube is given by A(s) = 6s. Substituting s(t) = 2+t into this equation, we get A(t) = 6(2+t) = 12+6t.

Part C: To find the time when the surface area equals 216 square inches, we set A(t) = 216 and solve for t. 12+6t = 216 ⇒ 6t = 204 ⇒ t = 34. Therefore, the surface area will equal 216 square inches after 34 hours.