Ms. Walker starts the fund with an initial deposit of $5, and the balance doubles each week. Thus, the balance of the fund after \( x \) weeks can be represented by the following equation:
\[ B(x) = 5 \cdot 2^x \]
where \( B(x) \) is the balance after \( x \) weeks.
We want to find the number of weeks \( x \) when the balance \( B(x) \) reaches $1,280. We can set up the equation like this:
\[ 5 \cdot 2^x = 1280 \]
To solve for \( x \), we first divide both sides by 5:
\[ 2^x = \frac{1280}{5} \]
Calculating \( \frac{1280}{5} \):
\[ 2^x = 256 \]
Next, we recognize that \( 256 \) can be expressed as a power of \( 2 \):
\[ 256 = 2^8 \]
So we have:
\[ 2^x = 2^8 \]
Since the bases are the same, we can equate the exponents:
\[ x = 8 \]
Therefore, it will take 8 weeks for the balance of the fund to reach $1,280.