MATH SOLVE

4 months ago

Q:
# A survey of 1,107 tourists visiting Orlando was taken. Of those surveyed:268 tourists had visited the Magic Kingdom258 tourists had visited Universal Studios68 tourists had visited both the Magic Kingdom and LEGOLAND79 tourists had visited both the Magic Kingdom and Universal Studios72 tourists had visited both LEGOLAND and Universal Studios36 tourists had visited all three theme parks58 tourists did not visit any of these theme parksHow many tourists only visited the LEGOLAND (of these three)?

Accepted Solution

A:

Answer:602 tourists visited only the LEGOLAND.Step-by-step explanation:To solve this problem, we must build the Venn's Diagram of this set.I am going to say that:-The set A represents the tourists that visited LEGOLAND-The set B represents the tourists that visited Universal Studios-The set C represents the tourists that visited Magic Kingdown.-The value d is the number of tourists that did not visit any of these parks, so: [tex]d = 58[/tex]We have that:[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]In which a is the number of tourists that only visited LEGOLAND, [tex]A \cap B[/tex] is the number of tourists that visited both LEGOLAND and Universal Studies, [tex]A \cap C[/tex] is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and [tex]A \cap B \cap C[/tex] is the number of students that visited all these parks.By the same logic, we have:[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex][tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]This diagram has the following subsets:[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]There were 1,107 tourists suveyed. This means that:[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]We start finding the values from the intersection of three sets.The problem states that:36 tourists had visited all three theme parks. So:[tex](A \cap B \cap C) = 36[/tex]72 tourists had visited both LEGOLAND and Universal Studios. So:[tex](A \cap B) + (A \cap B \cap C) = 72[/tex][tex](A \cap B) = 72 - 36[/tex][tex](A \cap B) = 36[/tex]79 tourists had visited both the Magic Kingdom and Universal Studios[tex](B \cap C) + (A \cap B \cap C) = 79[/tex][tex](B \cap C) = 79 - 36[/tex][tex](B \cap C) = 43[/tex]68 tourists had visited both the Magic Kingdom and LEGOLAND[tex](A \cap C) + (A \cap B \cap C) = 68[/tex][tex](A \cap C) = 68 - 36[/tex][tex](A \cap C) = 32[/tex]258 tourists had visited Universal Studios:[tex]B = 258[/tex][tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex][tex]258 = b + 43 + 36 + 36[/tex][tex]b = 143[/tex]268 tourists had visited the Magic Kingdom:[tex]C = 268[/tex][tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex][tex]268 = c + 32 + 43 + 36[/tex][tex]c = 157[/tex]How many tourists only visited the LEGOLAND (of these three)?We have to find the value of a, and we can do this by the following equation:[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex][tex]a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107[/tex][tex]a = 602[/tex]602 tourists visited only the LEGOLAND.