Let's carefully analyze and solve this problem step by step.
Ms. Wakker's class is trying to raise [tex]$1,280 for a field trip. She starts the fund with an initial deposit of $[/tex]5, and each week the balance of the fund doubles.
1. Understanding the growth pattern:
The balance of the fund after each week is double the balance of the previous week. This is a classic example of exponential growth. Let's represent the balance after [tex]\( x \)[/tex] weeks with [tex]\( B \)[/tex]. Initially, [tex]\( B = 5 \)[/tex].
After 1 week, the balance is [tex]\( B = 5 \times 2 \)[/tex].
After 2 weeks, the balance is [tex]\( B = 5 \times 2^2 \)[/tex].
And so forth.
Therefore, the relationship between the balance [tex]\( B \)[/tex] and the number of weeks [tex]\( x \)[/tex] can be generalized by the equation:
[tex]\[
B = 5 \times 2^x
\][/tex]
2. Setting up the equation:
We need to find the number of weeks [tex]\( x \)[/tex] it will take for the balance to reach \[tex]$1,280. Thus, we set up the equation where \( B = 1280 \):
\[
5 \times 2^x = 1280
\]
3. Solving for \( x \):
To isolate \( x \), we first divide both sides of the equation by 5:
\[
2^x = \frac{1280}{5}
\]
Simplifying the right-hand side:
\[
2^x = 256
\]
Now, we need to determine the power to which 2 must be raised to get 256. We recognize that:
\[
256 = 2^8
\]
Thus:
\[
x = 8
\]
4. Conclusion:
The correct equation to model the growth of the fund and the number of weeks needed to reach $[/tex]1,280 is:
[tex]\[
5 \times 2^x = 1280
\][/tex]
And it takes:
[tex]\[
x = 8 \text{ weeks}
\][/tex]
Given the options in the question, the correct answer is:
D. [tex]\( 5(2)^x=1,280 ; x=8 \)[/tex]
