To understand Mr. Hibbard's solution regarding the magic penny that doubles in value every day, we can break down the problem:
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The expression for the value of the penny is given as: \[ V = 1 \times 2^x \] where \( x \) is the number of days since Mr. Hibbard found the penny.
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The problem states that after 14 days, the value has reached $163.84. We will calculate this using the correct substitution into the expression.
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Substituting \( x = 14 \) into the expression: \[ V = 1 \times 2^{14} \]
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Now, we calculate \( 2^{14} \): \[ 2^{14} = 16384 \]
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Since the calculation says that the penny's value is $163.84, we need to consider that the expression we actually want is: \[ V = 0.01 \times 2^{14} \]
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Here \( 0.01 \) represents the initial value of the penny in dollars (or 1 cent).
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Now substituting \( x = 14 \): \[ V = 0.01 \times 2^{14} = 0.01 \times 16384 = 163.84 \]
So, Mr. Hibbard's correct approach was to substitute \( x = 14 \) into the expression \( 0.01 \times 2^{14} = 163.84 \), which correctly calculates the value of the penny after 14 days.
None of the incorrect answers provided (such as substituting \( x = 0 \) or \( x = 2 \)) lead to the correct value of $163.84. The correct substitution at day 14 using \( 0.01 \) as the initial value is how he reached the solution.
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