39. For what value of t is 100(1.1)^t = 1102? Give your answer as a fraction.
Answer: A
t = 25
To solve the equation 100(1.1)^t = 1102, we can first divide both sides by 100, yielding (1.1)^t = 11.02. By taking the logarithm of both sides, we can isolate t and find that t equals 25.
A) 25
This option is correct because substituting t = 25 into the original equation gives us 100(1.1)^25, which simplifies to approximately 1102. Thus, this value satisfies the equation perfectly.
B) 26
This option is incorrect. If we substitute t = 26 into the equation, we get 100(1.1)^26, which results in a value greater than 1102, thus failing to satisfy the original equation.
C) 20
This option is also incorrect. Substituting t = 20 results in 100(1.1)^20, which is significantly less than 1102. Therefore, this value does not satisfy the equation.
D) 24
This option is incorrect as well. If we substitute t = 24 into the equation, we find that 100(1.1)^24 still results in a value below 1102, indicating that this choice does not satisfy the equation.
Conclusion
The correct value of t is 25, as it is the only option that satisfies the equation 100(1.1)^t = 1102. All other options fail to meet this requirement, either exceeding or falling short of the target value.