Equations, expressions, and inequalitie #6 - Questions and Answers

Question 1 A disc jockey has to choose three songs for the last few minutes of his evening show. If there are nine songs that he feels are appropriate for that time slot, then how many ways can he choose and arrange to play three of those nine songs? Answer Assuming he doesn't want to repeat any songs, then for the first song, he can choose any of the nine. For the second he can choose any of the nine except the one he played first. So for each of those original nine songs, their are eight options for the second. Thus there are 8x9 = 72 ways to play the first two. One the first two are chosen, there are seven songs from which he can choose for the last. So for each of the 72 options for the first two songs, there are seven options for the last song. Thus there are 7×8×9 = 504 total ways he could choose and arrange the three songs from the nine. Question 2 Two spelunkers (cave explorers) were exploring different branches of an underground cavern. The first was able to descend 198 ft farther than twice the second. If the first spelunker descended a 1218 ft, how far was the second spelunker able to descend? Answer Let x be the distance the second spelunker descended so that the first spelunker descended 2x + 198, both in ft. 2x + 198 = 1218 2x = 1020 x = 510 The second spelunker was able to descend a distance of 510 feet. Question 3 A standard piece of paper is 8.5 inches by 11 inches. A piece of legal-size paper is 8.5 inches by 14 inches. By what scale factor k would you need to dilate the standard paper so that you could fit two pages on a single piece of legal paper? Answer You would need the breadth of the length of the legal paper to match the width of the ordinary paper. The ratio of the width of the legal paper to the length of the standard paper must therefore be the scaling factor: k = 8.5 / 11 = 17 / 22 We multiply the scale factor by two times the width of the standard paper to see if the widths of the two dilated standard papers fit along the length of the legal paper: (17/22) * (2 * 8.5) ~ 13.1 < 14 So, the two pages will fit for this value of k. Question 4 In January, the average high temperature for Calgary is 9.9°C greater than Winnipeg’s average high temperature. The sum of these two temperatures is -15.5°C. What is the average high temperature in January for each of these two cities? Answer c = calgary w = winnipeg The two equations would be: c = w + 9.9 (as the weather in Calgary is 9.9 degrees warmer than Winnipeg) and -15.5 = w + c (as the sum of the two temperatures is -15.5 degrees) We already have the C value (C = w + 9.9) so we can plug that into the other equation, as -15.5 = w + (w + 9.9), which would be: -15.5 = 2w + 9.9 simplified, then -25.4 = 2w and finally w = -12.7. Putting that value into the other equation as -15.5 = -12.7 + c, -15.5 + 12.7 = -2.8, therefore c = -2.8 Answer: w = -12.7c = -2.8 Question 5 Suppose you ask a friend to randomly choose an integer between 1 and 10, inclusive. What is the probability that the number will be more than 4 or odd? Answer Of the ten integers between one and ten inclusive, six are more than four (that is, 5,6,7,8,9,10). The probability is calculated by dividing the number of likely outcomes by the total number of outcomes: P(> 4) = # of favorable outcomes / # of possible outcomes = 6 / 10 5 of the 10 integers between 1 and 10 inclusive are odd (that is, 1,3,5,7,9). P(odd) = # of favorable outcomes / # of possible outcomes = 5 / 10 Three of the ten integers, ranging from one to ten, are more than four and odd (that is, 5,7,9). P(> 4 and odd) = # of favorable outcomes / # of possible outcomes = 3/10 If there are any two occurrences, apply the general addition rule: P(A∪B) = P(A) + P(B) – P(A∩B) P(> 4 or odd) = P(> 4) + P(odd) - P(> 4 and odd) = 6/10 + 5/10 - 3/10 = 8/10 = 4/5