Aand b are involved in a duel. the rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. if one or both are hit, then the duel is over. if both shots miss, then they repeat the process. suppose that the results of the shots are independent and that each shot of a will hit b with probability pa, and each shot of b will hit a with probability pb. what is (a) the probability that a is not hit? (b) the probability that both duelists are hit? (c) the probability that the duel ends after the nth round of shots? (d) the conditional probability that the duel ends after the nth round of shots given that a is not hit? (e) the conditional probability that the duel ends after the nth round of shots given that both duelists are hit?

Brittany has designed four different birdcages, each with the same volume: a square pyramid, a cylinder, a cone, and a sphere. the zoo wants to use one of brittany’s designs but plans to change the volume of the cage. find the design change that would result in the birdcage with the greatest volume. increase the area of the base of the square pyramid by a factor of ten. increase the height of the cylinder by a factor of ten. increase the radius of the cone by a factor of ten. increase the radius of a sphere by a factor of ten. juan wants to build a greenhouse in his back yard. he needs to maximize the ratio of floor area to structure volume in order to provide the most room for plants while minimizing the area to be heated. find the description of the structure that best meets juan’s criteria. a cube with a side length of 12 feet a cone with a diameter of 12 feet and a height of 10 feet a square pyramid with a base side length of 12 feet and a height of 9 feet a hemisphere with a diameter of 12 feet marion has a bicycle that she is planning to take for a ride. the rim of the rear wheel lies 41 cm from the center of the wheel. the tire itself has a thickness of 3 cm. in order to marion to ride her bicycle 96,712 cm, approximately how many rotations will the rear wheel make? 350 15.9 376 5133 an isosceles trapezoid has perimeter of 42 inches. each of the congruent nonparallel sides is 5 inches long, and the trapezoid is 3 inches tall. how long are the two parallel sides? 10 inches and 16 inches 12 inches and 20 inches 16 inches and 16 inches 10 inches and 22 inches

The rectangle on the right is the image of the rectangle on the left. what transformation would result in this image? reflection rotation translation dilation

Find the value of x. give reasons to justify your solutions! c ∈ ae