Final answer:
The marching band with 75 members can form two rectangular formations by arranging themselves into rectangles with 5 rows of 15 members each, or 15 rows of 5 members each. This takes into account that the factors of 75 should yield whole numbers for both the 6-row and 9-row formations.
Step-by-step explanation:
To figure out how a marching band with 75 members can form two different rectangular formations, one with 6 rows and another with 9 rows, we need to understand the concept of factors and multiples. Since the total number of band members is 75, each rectangle formation must be organized such that the product of its dimensions equals 75. This is because the number of members in each row multiplied by the number of rows should give us the total number of members in the formation.
For the rectangle with 6 rows, we divide 75 by 6 to determine how many members will be in each row: 75 ÷ 6 = 12.5. Since we cannot have half a member, this formation is not possible. Now, let's consider the rectangle with 9 rows. We divide 75 by 9 to get the number of members per row: 75 ÷ 9 = 8.33, which is also not possible for the same reason.
For both formations to have an integer number of band members per row, we must find factors of 75 that yield whole numbers when divided. The appropriate factors of 75 are 1, 3, 5, 15, 25, and 75. From these, the only pair of factors that fit our formations of 6 and 9 rows are 5 and 15. Therefore, the band members can form a rectangle with 5 rows of 15 members each (5 x 15 = 75) and another rectangle with 15 rows of 5 members each (15 x 5 = 75).