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Answer :
To solve the question, we need to determine how long it will take for the balance of Sylvia's loan to reach specific amounts given in the problem.
1. Understanding the Loan Balance Equation:
The loan balance is modeled by the equation:
[tex]\[
y = -382.05x + 25,077.40
\][/tex]
Here, [tex]\(y\)[/tex] is the loan balance, and [tex]\(x\)[/tex] is the number of months.
2. Determining the Balance at Different Months:
a. For a Balance of \[tex]$17,000:
We want to find when the balance will be \$[/tex]17,000. Plug \[tex]$17,000 into the equation for \(y\):
\[
17,000 = -382.05x + 25,077.40
\]
Solving for \(x\):
\[
25,077.40 - 17,000 = 382.05x
\]
\[
8,077.40 = 382.05x
\]
\[
x \approx \frac{8,077.40}{382.05} \approx 21.14
\]
Since \(x\) must be a whole number (as it represents months), we round 21.14 to 21. So, the balance will be approximately \$[/tex]17,000 around month 21.
b. For a Balance of \[tex]$0:
To find out when the loan will be fully paid off (balance of \$[/tex]0), set [tex]\(y\)[/tex] to 0:
[tex]\[
0 = -382.05x + 25,077.40
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
25,077.40 = 382.05x
\][/tex]
[tex]\[
x \approx \frac{25,077.40}{382.05} \approx 65.44
\][/tex]
Rounding 65.44, we find that the loan will be paid off after approximately 66 months.
3. Selecting the Correct Answers:
- For approximately \[tex]$17,000, the loan balance is around month 21.
- The loan reaches a balance of \$[/tex]0 around month 66.
Therefore, the filled options should be:
- The balance of the loan will be approximately \[tex]$17,000: month 21
- The balance of the loan will reach \$[/tex]0: month 66
1. Understanding the Loan Balance Equation:
The loan balance is modeled by the equation:
[tex]\[
y = -382.05x + 25,077.40
\][/tex]
Here, [tex]\(y\)[/tex] is the loan balance, and [tex]\(x\)[/tex] is the number of months.
2. Determining the Balance at Different Months:
a. For a Balance of \[tex]$17,000:
We want to find when the balance will be \$[/tex]17,000. Plug \[tex]$17,000 into the equation for \(y\):
\[
17,000 = -382.05x + 25,077.40
\]
Solving for \(x\):
\[
25,077.40 - 17,000 = 382.05x
\]
\[
8,077.40 = 382.05x
\]
\[
x \approx \frac{8,077.40}{382.05} \approx 21.14
\]
Since \(x\) must be a whole number (as it represents months), we round 21.14 to 21. So, the balance will be approximately \$[/tex]17,000 around month 21.
b. For a Balance of \[tex]$0:
To find out when the loan will be fully paid off (balance of \$[/tex]0), set [tex]\(y\)[/tex] to 0:
[tex]\[
0 = -382.05x + 25,077.40
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
25,077.40 = 382.05x
\][/tex]
[tex]\[
x \approx \frac{25,077.40}{382.05} \approx 65.44
\][/tex]
Rounding 65.44, we find that the loan will be paid off after approximately 66 months.
3. Selecting the Correct Answers:
- For approximately \[tex]$17,000, the loan balance is around month 21.
- The loan reaches a balance of \$[/tex]0 around month 66.
Therefore, the filled options should be:
- The balance of the loan will be approximately \[tex]$17,000: month 21
- The balance of the loan will reach \$[/tex]0: month 66
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