Question

Mr. Hibbard found a magic penny which doubles in value every day. The expression .01 ⋅ 2^x  can be used to find the value of his coin 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.
A) Mr. Hibbard substituted x=14 into the expression .01 • 2^14 = 163.48.
B) Mr. Hibbard substituted x=14 into the expression .01 • 2 • 14 = 163.48
C) Mr. Hibbard substituted x=2 into the expression .01 • 2 • 2 = 163.48.
D) Mr. Hibbard substituted x=2 into the expression .01 • 2^2 = 163.48.

Answer 1

To find the correct solution, let's evaluate the expression \( .01 \cdot 2^x \) for the value of \( x \) that corresponds to 2 weeks, which is 14 days.

Substituting \( x = 14 \) into the expression, we have:

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

Calculating \( 2^{14} \):

\[
2^{14} = 16384
\]

Now, multiplying by \( 0.01 \):

\[
0.01 \cdot 16384 = 163.84
\]

This matches the final value that Mr. Hibbard found.

Now, let's analyze the answer choices provided:

A) Mr. Hibbard substituted \( x=14 \) into the expression \( 0.01 \cdot 2^{14} = 163.84 \) (Correct)

B) Mr. Hibbard substituted \( x=14 \) into the expression \( 0.01 \cdot 2 \cdot 14 = 163.48 \) (Incorrect)

C) Mr. Hibbard substituted \( x=2 \) into the expression \( 0.01 \cdot 2 \cdot 2 = 163.48 \) (Incorrect)

D) Mr. Hibbard substituted \( x=2 \) into the expression \( 0.01 \cdot 2^{2} = 163.48 \) (Incorrect)

Thus, the correct answer is:

**A) Mr. Hibbard substituted x=14 into the expression .01 • 2^14 = 163.84.**