Problem 7: 1998-1999

A Rubikk's Cyoob (to avoid copyright infringements) is accidentally dropped into a small pile of nuclear waste, causing two major results. First, the edges of the cube begin to expand at the rate of two feet per second. Second, the cyoob has increased crime fighter capabilities, including but not limited to the power to fly, fill out tax forms correctly, and chew through rock.

(a)    What is the rate of change of the surface area of of the cyoob when the edges are 12 feet long? 210 feet long?

(b)    What is the rate of change of the volume of the cube when the edges are 12 and 210 feet long?

(c)    What would be the name of the super villain the cyoob would fight?

Solution:
(a)    First, we need the equation for surface area of a cube, and we'll take the derivative of it with respect to time. If you think about it, the surface area of a cube is given by s = 6e2, since each face has an area of e2, where e is the length of an edge. Now, we know that de/dt = 2 ft/sec, as given in the problem, and we want to find ds/dt. We take the derivative with respect to time to get the following:
Notice that I plugged in 2 for de/dt. Now, we can find the two answers asked by part (a) by plugging them in, separately, for e:
    *    When e = 12, ds/st = 24(12) = 288 ft2/sec.
    *    When e = 210, ds/dt = 5040 ft2/sec.
Don't forget units; it's unnecessary to carry them all the way through, but at least insert them in the end. I use ft2/sec since area must be in square feet, not just feet.

(b)    We follow a method similar to part (a), but use the volume formula:

Therefore, we get answers of 864 ft3/sec and 264,600 ft3/sec, respectively, plugging back in as we did in part (a).

(c)    The major super villain the cyoob would fight is a villain called The End of the Eighties.


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