What is the relationship among the length L, width W and diagonal (let's call the diagonal Z) of a rectangle?

How many of the tiles on this floor would it take to cover, as nearly as possible, a hemispherical dome 600 feet in diameter?  What percent of the area of the dome would be left uncovered because of the cracks between the tiles?

How many corners are there in an n x n grid of square tiles?  Why can't the number of possible corner-to-corner lengths exceed this number?  How many reasons can you think of why the number of lengths is less than this number?  How much does each reason reduce the number?

For a general exponential graph, if you use equally-spaced short intervals, the ratios of the slopes will be constant, to the extent that your intervals are short.  For a quadratic graph it is the slopes themselves that change at a constant rate.

Velocity is the average rate of change of position with respect to clock time.  Acceleration is the average rate of change of velocity with respect to clock time.

Q: How do I know if my estimates of water depth are high or low? 

A: Suppose you are estimating the amount of depth change for a certain time interval, based on the rate at the beginning of the interval.  If the rate of depth change actually gets smaller during that interval, then did you overestimate or underestimate?

Q:  If the diagonal of a square is 7, then if a and b are the sides I know that a^2 + b^2 = 7^2 so a^2 + b^2 = 49.  Not sure where to go from there.

A:  The sides of a square are equal.  So instead of a and b, you could use s for both.  Then you could solve the equation.

Q:  How to fit tiles with sides 7/6 and 5/8 units to form a rectangle we can cover with 1 ft square tiles?

A:  Try a simpler problem.  You are going to build this rectangle by first making a row.  How many 7/6 unit tiles do you need to make a row whose length can be matched by a row of 1 ft tiles? 

There is no limit to how much pressure can increase as the depth of an incompressible fluid increases.  There is a limit to the pressure at which any given fluid remains incompressible, but for most fluids (water, magma, melted metals) that limit is very, very high.

A slope represents an average rate and an average rate can represent a slope.  The difference is interpretation.  You have to know how to translate one into the other, for the given situation.

A cube 1.5 units on a side can be subdivided into cubes .5 units on a side.  How many .5-unit cubes are required to make a 1.5-unit cube?  How many .5-unit cubes does it take to make a 1-unit cube?  How many times more stuff do you therefore require to fill a 1.5-unit cube, compare to a 1-unit cube?

If you're making $2000/month after 5 months, and $3000/month after 15 months, then as a first estimate it would be reasonable to assume that you're making $2500/month on the average.  Then if you got more information you could modify your estimate accordingly.