Question
Mr. Hibbard found a magic penny which doubles in value every day. The expression 1 * 2 ^ x where x is the number of days since Mr. Hibbard found the penny Mr. Hibbard realized after just 14 days, the penny had become $163.841 How did he reach this solution? point) Mr. Hibbard substituted x = 0 into the expression 0.01 * 2 ^ 2 = 163.48 Mr. Hibbard substituted x = 2 into the expression 0.01 * 2 * 2 = 163.48 Hibbard substituted x = 14 into the expression.01.21 0.01 * 2 ^ 14 = 163.48 Mr. Hibbard substituted x = 14 into the expression 01*2*14***163.48
Answers
GPT-4o mini
To understand Mr. Hibbard's solution regarding the magic penny that doubles in value every day, we can break down the problem:
1. 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.
2. The problem states that after 14 days, the value has reached $163.84. We will calculate this using the correct substitution into the expression.
3. Substituting \( x = 14 \) into the expression:
\[
V = 1 \times 2^{14}
\]
4. Now, we calculate \( 2^{14} \):
\[
2^{14} = 16384
\]
5. Since the calculation says that the penny's value is $163.84, we need to consider that the expression we actually want is:
\[
V = 0.01 \times 2^{14}
\]
6. Here \( 0.01 \) represents the initial value of the penny in dollars (or 1 cent).
7. 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.
1. 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.
2. The problem states that after 14 days, the value has reached $163.84. We will calculate this using the correct substitution into the expression.
3. Substituting \( x = 14 \) into the expression:
\[
V = 1 \times 2^{14}
\]
4. Now, we calculate \( 2^{14} \):
\[
2^{14} = 16384
\]
5. Since the calculation says that the penny's value is $163.84, we need to consider that the expression we actually want is:
\[
V = 0.01 \times 2^{14}
\]
6. Here \( 0.01 \) represents the initial value of the penny in dollars (or 1 cent).
7. 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.