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Sagot :
To determine the number of weeks it will take for the balance of the fund to reach \[tex]$1,280, we need to set up the correct equation and then solve for \( x \).
1. Identifying the initial amount and growth rate:
- The initial amount, \( S \), is \$[/tex]5.
- Each week, the balance is doubled, meaning it's multiplied by 2.
2. Setting up the correct exponential equation:
- We know that the fund balance grows exponentially as [tex]\( S \cdot 2^x \)[/tex], where [tex]\( S \)[/tex] is the starting amount, and [tex]\( x \)[/tex] is the number of weeks.
- Given that the goal is \[tex]$1,280, the equation we need is: \[ 5 \cdot 2^x = 1,280 \] 3. Solving for \( x \): - To solve for \( x \), we need to isolate \( x \) in the equation. - Divide both sides of the equation by 5 to simplify: \[ 2^x = \frac{1,280}{5} \] \[ 2^x = 256 \] - Now, we identify the power of 2 that equals 256: \[ 2^x = 2^8 \] \[ x = 8 \] Thus, the equation that represents the situation correctly is: \[ 5 \cdot 2^x = 1,280 \] And it will take 8 weeks for the balance to reach \$[/tex]1,280.
Therefore, the correct answer is:
C. [tex]\( 5(2)^x = 1,280 ; x = 8 \)[/tex]
- Each week, the balance is doubled, meaning it's multiplied by 2.
2. Setting up the correct exponential equation:
- We know that the fund balance grows exponentially as [tex]\( S \cdot 2^x \)[/tex], where [tex]\( S \)[/tex] is the starting amount, and [tex]\( x \)[/tex] is the number of weeks.
- Given that the goal is \[tex]$1,280, the equation we need is: \[ 5 \cdot 2^x = 1,280 \] 3. Solving for \( x \): - To solve for \( x \), we need to isolate \( x \) in the equation. - Divide both sides of the equation by 5 to simplify: \[ 2^x = \frac{1,280}{5} \] \[ 2^x = 256 \] - Now, we identify the power of 2 that equals 256: \[ 2^x = 2^8 \] \[ x = 8 \] Thus, the equation that represents the situation correctly is: \[ 5 \cdot 2^x = 1,280 \] And it will take 8 weeks for the balance to reach \$[/tex]1,280.
Therefore, the correct answer is:
C. [tex]\( 5(2)^x = 1,280 ; x = 8 \)[/tex]
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