Answer :
To find the linear model for the simple interest on the loan after [tex]\( t \)[/tex] years, let's follow these steps:
1. Understand the Given Information:
- Initial loan amount (Principal, [tex]\( P \)[/tex]): \[tex]$7,000
- Final amount after 2 years: \$[/tex]7,140
- Time period ([tex]\( t \)[/tex]): 2 years
2. Calculate the Interest Earned:
- Interest earned is the increase in the amount after 2 years.
[tex]\[ \text{Interest Earned} = \text{Final Amount} - \text{Initial Amount} = 7140 - 7000 = 140 \][/tex]
3. Determine the Simple Interest Rate:
- Simple interest is calculated using the formula:
[tex]\[ \text{Interest} = P \times r \times t \][/tex]
- Here, [tex]\( r \)[/tex] is the interest rate we need to find. Rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\text{Interest}}{P \times t} \][/tex]
- Substitute the known values:
[tex]\[ r = \frac{140}{7000 \times 2} = \frac{140}{14000} = 0.01 \][/tex]
4. Construct the Linear Model:
- The linear model for simple interest over [tex]\( t \)[/tex] years can be expressed as:
[tex]\[ A(t) = P \times r \times t \][/tex]
- Substitute the known values:
[tex]\[ A(t) = 7000 \times 0.01 \times t \][/tex]
Therefore, the correct linear model for the simple interest on this loan after [tex]\( t \)[/tex] years is:
[tex]\[ A(t) = 7000 \cdot 0.01 \cdot t \][/tex]
This matches the first option in the list of choices provided.
1. Understand the Given Information:
- Initial loan amount (Principal, [tex]\( P \)[/tex]): \[tex]$7,000
- Final amount after 2 years: \$[/tex]7,140
- Time period ([tex]\( t \)[/tex]): 2 years
2. Calculate the Interest Earned:
- Interest earned is the increase in the amount after 2 years.
[tex]\[ \text{Interest Earned} = \text{Final Amount} - \text{Initial Amount} = 7140 - 7000 = 140 \][/tex]
3. Determine the Simple Interest Rate:
- Simple interest is calculated using the formula:
[tex]\[ \text{Interest} = P \times r \times t \][/tex]
- Here, [tex]\( r \)[/tex] is the interest rate we need to find. Rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\text{Interest}}{P \times t} \][/tex]
- Substitute the known values:
[tex]\[ r = \frac{140}{7000 \times 2} = \frac{140}{14000} = 0.01 \][/tex]
4. Construct the Linear Model:
- The linear model for simple interest over [tex]\( t \)[/tex] years can be expressed as:
[tex]\[ A(t) = P \times r \times t \][/tex]
- Substitute the known values:
[tex]\[ A(t) = 7000 \times 0.01 \times t \][/tex]
Therefore, the correct linear model for the simple interest on this loan after [tex]\( t \)[/tex] years is:
[tex]\[ A(t) = 7000 \cdot 0.01 \cdot t \][/tex]
This matches the first option in the list of choices provided.
