I send you 1,000,000 dice. Each die is ½ inch, wide, tall and deep.

Part 1. If the dice come in a package, neatly arranged in one big cube, How big is the cube?
Would it fit under a table?
Would you need a garage to fit the package?
Or would you need a warehouse.

Part 2. If you arranged the dice into a square, one layer deep, how much space would you need?
Your living room floor?
A basketball court?
A huge parking lot?

Part 3. Stack the 1,000,000 dice into a giant tower, just one cube thick. How high will the pile be?
As high as a house? (25 feet)
As high as the empire state building in New York city? (1,472 feet)
Higher than Mount Everest (29,000 feet)

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

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