9. If 3 < a < 7 < b, which of the following must be greater than 20?
Answer: C
ab must be greater than 20.
Given that 3 < a < 7 and 7 < b, we can establish that the minimum value for a is just above 3 and the minimum value for b is just above 7. Therefore, when we calculate the product ab, it will yield a value that is greater than 3 * 7, which equals 21.
A) a²
The expression a² ranges from just above 9 (since a is slightly more than 3) to just below 49 (since a is slightly less than 7). Therefore, a² can be less than 20 when a is close to 3, which disqualifies this option as a must.
B) 2b
The expression 2b starts from just above 14 (since b is slightly more than 7) and can go higher. However, since 2b could be less than 20 if b is just over 7, this option does not necessarily satisfy the condition.
C) ab
This expression is determined to be greater than 21 since the minimum values for a and b are just above 3 and 7, respectively. Therefore, ab is definitely greater than 20, making this the only option that must be greater than 20.
D) b + a
The sum b + a starts from just above 10 (since a is slightly more than 3 and b is slightly more than 7). Thus, it can be less than 20 if a is close to 3, which makes this option incorrect.
Conclusion
The expression ab is the only option that must be greater than 20, as it consistently exceeds the product of the minimum values of a and b. All other options can potentially yield values less than 20 depending on the specific values of a and b within their defined ranges. Thus, option C is definitively correct.