10. Let a > 0, b > 0, and 2 > 3b. If (2a + 3b) : (2a - 3b) = 7 : 5, then which of the following is equivalent to (4a² + 9b²) : (4a² - 9b²)?
Answer: E
(4a² + 9b²) : (4a² - 9b²) is equivalent to 49:25
Given the relationship (2a + 3b) : (2a - 3b) = 7 : 5, we can derive the equivalent ratio for (4a² + 9b²) : (4a² - 9b²). Through algebraic manipulation, we find that this ratio simplifies to 49:25.
A) 3:5
This option is incorrect as it does not follow from the derived ratio. The ratio 3:5 corresponds to a different relationship and does not satisfy the condition provided in the question.
B) 55:53:00
This option is not relevant as it presents an unusual format for a ratio. The correct form of the ratio must maintain whole number proportions, which 55:53:00 does not.
C) 37:35:00
Similar to option B, this ratio is presented in an unconventional format and does not represent a simple ratio. The correct response must align with the established ratios derived from the initial condition.
D) 5:3
This option is incorrect as it significantly deviates from the ratio derived from the problem statement. The ratio of 5:3 does not satisfy the relationships outlined in the original equation.
E) 49:25
This option is correct as it directly corresponds to the correctly derived ratio from the expression (4a² + 9b²) : (4a² - 9b²). It maintains the necessary proportions established by the earlier provided ratio of (2a + 3b) : (2a - 3b).
Conclusion
The correct answer is 49:25, as it accurately reflects the derived result from the initial condition of the problem. All other options either misrepresent the ratio format or do not align with the mathematical derivations necessary to solve the problem. Thus, option E stands as the only valid choice.