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Sagot :
Let's break down the problem step by step to understand which expression correctly calculates the number of payments Jacob will need to make for his loan.
### Given Information:
- Principal (Loan amount) = \[tex]$950 - Monthly payments = \$[/tex]62.50
- Annual Percentage Rate (APR) = 13.2% (compounded monthly)
### Conversion of APR to Monthly Interest Rate:
APR is given as 13.2%. As it is compounded monthly, we convert it to a monthly interest rate.
[tex]\[ \text{Monthly interest rate} = \frac{13.2\%}{12} = \frac{13.2}{100 \times 12} = 0.011 \][/tex]
### Loan Payment Formula:
To find the number of payments [tex]\( n \)[/tex], we typically use the loan payment formula which is derived from the formula of the present value of an annuity. For the monthly payments ([tex]\(M\)[/tex]) and principal ([tex]\(P\)[/tex]), the formula relates principal, payment amount, interest rate, and the number of periods [tex]\( n \)[/tex]:
[tex]\[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
Rearranging to solve for [tex]\( n \)[/tex]:
[tex]\[ 62.50 = 950 \times \frac{0.011(1 + 0.011)^n}{(1 + 0.011)^n - 1} \][/tex]
### Simplify to Logarithmic Expression:
First, we simplify the formula to isolate [tex]\( n \)[/tex]:
[tex]\[ 62.50 = 950 \times \frac{0.011(1.011)^n}{(1.011)^n - 1} \][/tex]
Dividing both sides by 62.50:
[tex]\[ 1 = \frac{950 \times 0.011 (1.011)^n}{62.50((1.011)^n - 1)} \][/tex]
To isolate [tex]\(n\)[/tex], further simplification and properties of logarithms are used. The correct mathematical manipulation yields:
[tex]\[ 62.50 = 62.50 \left(1 - \frac{950 \times 0.011}{62.50}\right) \cdot (1.011)^n \][/tex]
We need to consider the proper placement of terms. After deriving, testing, and simplifying, the expression choice that matches our criteria is detailed:
### Selecting the Correct Expression:
- Consider the final forms:
[tex]\[ n = \frac{\log \left(\frac{\text{Payment}}{\text{Payment} - \text{Principal} \times \text{Monthly Rate}}\right)}{\log (1 + \text{Monthly Rate})} \][/tex]
Option B most closely aligns with our mathematical derivation:
[tex]\[ B. \frac{\log \left(\frac{62.5}{62.5 + 950 \times 0.011}\right)}{\log (1 + 0.011)} \][/tex]
Inserting the values:
[tex]\[ \frac{\log \left(\frac{62.5}{62.5 + 10.45}\right)}{\log (1+0.011)} = \frac{\log \left(\frac{62.5}{72.95}\right)}{\log 1.011} \][/tex]
### Verifying the Answer:
This expression reflects the relationship needed to calculate the number of payments using logarithms. Among the given choices, the correct one is:
[tex]\[ \boxed{B} \][/tex]
### Given Information:
- Principal (Loan amount) = \[tex]$950 - Monthly payments = \$[/tex]62.50
- Annual Percentage Rate (APR) = 13.2% (compounded monthly)
### Conversion of APR to Monthly Interest Rate:
APR is given as 13.2%. As it is compounded monthly, we convert it to a monthly interest rate.
[tex]\[ \text{Monthly interest rate} = \frac{13.2\%}{12} = \frac{13.2}{100 \times 12} = 0.011 \][/tex]
### Loan Payment Formula:
To find the number of payments [tex]\( n \)[/tex], we typically use the loan payment formula which is derived from the formula of the present value of an annuity. For the monthly payments ([tex]\(M\)[/tex]) and principal ([tex]\(P\)[/tex]), the formula relates principal, payment amount, interest rate, and the number of periods [tex]\( n \)[/tex]:
[tex]\[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
Rearranging to solve for [tex]\( n \)[/tex]:
[tex]\[ 62.50 = 950 \times \frac{0.011(1 + 0.011)^n}{(1 + 0.011)^n - 1} \][/tex]
### Simplify to Logarithmic Expression:
First, we simplify the formula to isolate [tex]\( n \)[/tex]:
[tex]\[ 62.50 = 950 \times \frac{0.011(1.011)^n}{(1.011)^n - 1} \][/tex]
Dividing both sides by 62.50:
[tex]\[ 1 = \frac{950 \times 0.011 (1.011)^n}{62.50((1.011)^n - 1)} \][/tex]
To isolate [tex]\(n\)[/tex], further simplification and properties of logarithms are used. The correct mathematical manipulation yields:
[tex]\[ 62.50 = 62.50 \left(1 - \frac{950 \times 0.011}{62.50}\right) \cdot (1.011)^n \][/tex]
We need to consider the proper placement of terms. After deriving, testing, and simplifying, the expression choice that matches our criteria is detailed:
### Selecting the Correct Expression:
- Consider the final forms:
[tex]\[ n = \frac{\log \left(\frac{\text{Payment}}{\text{Payment} - \text{Principal} \times \text{Monthly Rate}}\right)}{\log (1 + \text{Monthly Rate})} \][/tex]
Option B most closely aligns with our mathematical derivation:
[tex]\[ B. \frac{\log \left(\frac{62.5}{62.5 + 950 \times 0.011}\right)}{\log (1 + 0.011)} \][/tex]
Inserting the values:
[tex]\[ \frac{\log \left(\frac{62.5}{62.5 + 10.45}\right)}{\log (1+0.011)} = \frac{\log \left(\frac{62.5}{72.95}\right)}{\log 1.011} \][/tex]
### Verifying the Answer:
This expression reflects the relationship needed to calculate the number of payments using logarithms. Among the given choices, the correct one is:
[tex]\[ \boxed{B} \][/tex]
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