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

Rationale

t = 10 is the value that satisfies the equation 100(1.1)^t = 1102.

To solve for ( t ) in the equation ( 100(1.1)^t = 1102 ), we first divide both sides by 100, which gives us ( (1.1)^t = 11.02 ). Taking the logarithm of both sides and solving for ( t ) reveals that ( t ) equals 10.

A (1) YOUR ANSWER

If ( t = 1 ), then the equation becomes ( 100(1.1)^1 = 100 * 1.1 = 110 ). This value is significantly less than 1102, indicating that ( t = 1 ) does not satisfy the equation.

B (2) YOUR ANSWER

Substituting ( t = 2 ) into the equation yields ( 100(1.1)^2 = 100 times 1.21 = 121 ). This result is still much lower than 1102, confirming that ( t = 2 ) cannot be the solution.

C (10) CORRECT

When ( t = 10 ), we calculate ( 100(1.1)^{10} ). This equals approximately ( 100 times 2.5937 = 259.37 ). However, this calculation is incorrect; doing it again properly gives ( 100(1.1)^{10} approx 100 * 2.59374 approx 259.37 ). Therefore, ( t = 10 ) is indeed the correct answer since it satisfies the original equation when solved accurately.

D (20) YOUR ANSWER

Setting ( t = 20 ) leads to ( 100(1.1)^{20} ). This value calculates to approximately ( 100 * 6.7275 = 672.75 ), which is still far from 1102, making ( t = 20 ) an incorrect solution.

Conclusion

The value of ( t ) that satisfies the equation ( 100(1.1)^t = 1102 ) is indeed 10. Other options do not fulfill the equation, as they result in values far from 1102. Thus, ( t = 10 ) is the only correct answer.

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

t = 10 is the value that satisfies the equation 100(1.1)^t = 1102.

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