How is the formula for the volume of a cylinder derived? drag and drop the correct word into each box to complete the explanation.

the rectangular prism and the cylinder have heights.

the area of the base of the rectangular prism and every cross section parallel to its base is πr2 .

the area of the base of the cylinder and every cross section parallel to its base is πr2 .

by cavalieri's principle, this rectangular prism and this cylinder have equal volumes.

the volume of the rectangular prism is πr2h .

therefore, the volume of the cylinder is

What can you determine about the solutions of this system

Atriangle has a side lengths of 18cm, 80 cm and 81cm. classify it as acute obtuse or right?

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.