Video Transcript
A metallic cube expands but preserves its shape as it is heated. What is the average rate of change of its surface area when its sides change from 69 centimeters to 69.7 centimeters?
We’re given a metallic cube with initial side length 69 centimeters. We’re told that when heated, the cube expands but preserves its shape and that the side length changes from 69 centimeters to 69.7 centimeters. And we’re asked for the average rate of change of its surface area during this expansion. There are two things we need to remind ourselves of to answer this question. The first is that the average rate of change of the function 𝑓 of 𝑥 when 𝑥 changes from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏 is 𝑓 evaluated at 𝑥 is equal to 𝑏 minus 𝑓 evaluated at 𝑥 is equal to 𝑎 over 𝑏 minus 𝑎. And the second is the surface area of a cube.
Remember, a cube has six faces. And if it has side length 𝑥, then the area of one side is 𝑥 squared. The surface area of the whole cube is therefore six 𝑥 squared. And because we want to find the average rate of change of the surface area, this is our function 𝑓 of 𝑥. So the surface area of a cube with side length 𝑥 is six 𝑥 squared. Now going back to our average rate of change of the surface area, our cube starts with a side length of 𝑥 is 69 centimeters so that in our average rate of change, 𝑎 is equal to 69. Our side length expands to 69.7, so 𝑏 is 69.7.
So let’s insert these into our average rate of change function. The average rate of change is 𝑓 evaluated at 69.7 minus 𝑓 evaluated at 69 divided by 69.7 minus 69. To evaluate this, we need to substitute 𝑥 is 69.7 into our function 𝑓 of 𝑥 is six 𝑥 squared and do the same for 𝑥 is 69. And 𝑓 evaluated at 𝑥 at 69.7 is six times 69.7 squared, which is 29148.54. And 𝑓 evaluated at 𝑥 is 69 is six times 69 squared, and that’s 28566. Substituting these into our average rate of change gives us an average rate of change of 29148.54 minus 28566 divided by 69.7 minus 69, that is, 582.54 divided by 0.7, which evaluates to 832.2. And our units are centimeters squared per centimeter.
This corresponds to the change in area per unit length. The average rate of change of the surface area of a metallic cube where the sides expand from 69 to 69.7 centimeters is therefore 832.2 centimeters squared per centimeter.
