Hi Julie.

"Varies directly" means that as one quantity goes up, the other goes up proportionally. For example, if my gas mileage is 42 mpg, then I can say M = 42 x G, where the quantity M is the total miles and G is the total gallons. M varies directly as G. The constant 42 may be anything else, but the important idea is that if you increase the amount of fuel available (G), then M goes up.

"the square of" just means that the quantity is squared. For example, the distance in meters that a rock falls (D) varies directly as the square of the time (T) in seconds that it is freely falling in the relationship D = 4.9 T². So as the time goes up, the distance goes up faster and faster.

"Varies inversely" means that as one quantity goes up, the other quantity goes down to compensate. For example, when I run at the speed (S) from one end of a Canadian football field to the other, it takes time (T) in seconds. The higher my speed (S), the lower the time (T). In fact, S = 110 / T (Canadian football fields are 110 yards long), so we say that S varies inversely as T in this equation.

There's always some constant involved in these relationships (in my examples, these were 42, 4.9 and 110) and often we just call the constant k if we don't know it.

Now to your question:

X varies directly as the square of S and inversely as T.

So X = k S² / T.

Look at this equation (ignore k, it doesn't affect your answers).

Let's say S = 3, just for example. Then X = 9k / T. If we double S so S = 6 (this is your first question), then what does this do to the value of X?

Cheers,
Stephen La Rocque.