To understand how Mr. Hibbard reached the value of $163.84 after 2 weeks with the magic penny that doubles in value every day, we can break it down mathematically using the expression \(0.01 \times 2^x\), where \(x\) is the number of days since he found the penny.
Step-by-step Calculation:
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Determine the Number of Days:
Mr. Hibbard realized after just 2 weeks. Since there are 14 days in 2 weeks, we set \(x = 14\). -
Substitute \(x\) into the Expression:
Using the expression \(0.01 \times 2^x\), we substitute \(x\) with 14: \[ \text{Value after 14 days} = 0.01 \times 2^{14} \] -
Calculate \(2^{14}\):
First, we need to compute \(2^{14}\):- \(2^{10} = 1024\)
- Therefore, \(2^{14} = 2^{10} \times 2^4 = 1024 \times 16 = 16384\).
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Multiply by 0.01:
Now, we multiply this result by \(0.01\): \[ 0.01 \times 16384 = 163.84 \]
Conclusion:
So, after 14 days, the penny had indeed become $163.84. Mr. Hibbard used the expression correctly to reach this solution:
\[ \text{Value} = 0.01 \times 2^{14} = 163.84 \]
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