Question

Mr. Hibbard found a magic penny which doubles in value every day. The expression .01 ⋅ 2^x where x is the number of days since Mr. Hibbard found the penny. Mr. Hibbard realized after just 2 weeks the penny had become $163.84! How did he reach this solution

Answer 1

To solve the problem, we start with the expression for the value of the penny, which is given as:

\[
V = 0.01 \cdot 2^x
\]

where \( V \) is the value of the penny, and \( x \) is the number of days since Mr. Hibbard found it.

We know from the problem statement that after 2 weeks (which is 14 days), the value of the penny is $163.84. We can set up the equation using this information:

\[
0.01 \cdot 2^{14} = 163.84
\]

Now we can solve for \( 2^{14} \).

First, we divide both sides of the equation by 0.01:

\[
2^{14} = \frac{163.84}{0.01}
\]

Calculating the right side:

\[
163.84 \div 0.01 = 16384
\]

Now we have:

\[
2^{14} = 16384
\]

Next, we can verify that indeed \( 2^{14} = 16384 \). We can compute this directly:

- \( 2^1 = 2 \)
- \( 2^2 = 4 \)
- \( 2^3 = 8 \)
- \( 2^4 = 16 \)
- \( 2^5 = 32 \)
- \( 2^6 = 64 \)
- \( 2^7 = 128 \)
- \( 2^8 = 256 \)
- \( 2^9 = 512 \)
- \( 2^{10} = 1024 \)
- \( 2^{11} = 2048 \)
- \( 2^{12} = 4096 \)
- \( 2^{13} = 8192 \)
- \( 2^{14} = 16384 \)

Thus, \( 2^{14} = 16384 \) is confirmed to be correct.

By plugging the value back into the initial expression:

\[
0.01 \cdot 16384 = 163.84
\]

This confirms that after 14 days, the penny does indeed reach the value of $163.84. Therefore, Mr. Hibbard's calculations and conclusions about the value of the magic penny after 2 weeks are accurate.