For what value of t is 100(1.1)^t = 1102? Give your answer as a fraction.

Rationale

t = 25 is the solution to the equation 100(1.1)^t = 1102.

To solve for t, we can first divide both sides of the equation by 100, yielding (1.1)^t = 11.02. Taking the logarithm of both sides allows us to isolate t, leading to t = log(11.02) / log(1.1), which approximates to 25 when calculated.

A (25) CORRECT

This value of t accurately satisfies the original equation. By substituting back into the equation, 100(1.1)^25 equals approximately 1102, confirming that t = 25 is indeed correct.

B (26) YOUR ANSWER

If t were 26, substituting this value into the equation would yield 100(1.1)^26, which results in a value greater than 1102. Therefore, t = 26 does not satisfy the equation.

C (20) YOUR ANSWER

Substituting t = 20 into the original equation gives us 100(1.1)^20, which results in a value less than 1102. This indicates that t = 20 is too low to satisfy the equation.

D (24) YOUR ANSWER

For t = 24, the substitution into the equation yields 100(1.1)^24. While this value is closer to 1102 than t = 20 or 26, it still does not reach 1102, confirming that t = 24 is also incorrect.

Conclusion

The correct value of t that satisfies the equation 100(1.1)^t = 1102 is 25. By evaluating each choice, we see that t = 25 is the only option that correctly yields the target value when substituted back into the original equation, while all other options result in either too high or too low values.

For what value of t is 100(1.1)^t = 1102? Give your answer as a fraction.

t = 25 is the solution to the equation 100(1.1)^t = 1102.

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