If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume by a formula of the form
\(P=\frac{nRT}{V-nb}-\frac{an^2}{V^ 2}\)
(a) in which a, b, n, and R are constants. Find dP/dV.
(b) Now suppose temperature is not constant and that T, P, and V are functions of time t. Find dP/dt.
(c) Describe in words the meanings of dP/dV and dP/dt.

If the scale factor between two circles is 2x/5y what is the ratio of their areas?

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. which set of triangles shows the similar triangles created by the altitude in the diagram?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.