Aprinter is typesetting a book. he needs one piece of type for each digit in the page numbers of the book. each piece of type can be used only once. for example, page 3 and page 33 will require 3 pieces of type. how many twos will he need to number pages 1 through 232?

Abilliard ball is struck by a cue. it travels 1 0 0 cm 100cm before ricocheting off a rail and traveling another 1 2 0 cm 120cm into a corner pocket. the angle between the path as the ball approaches the rail and the path after it strikes the rail is 4 5 ∘ 45 ∘ . how far is the corner pocket from where the cue initially struck the ball? do not round during your calculations. round your final answer to the nearest centimeter.