Answer :
To determine which statement accurately represents the given data and the models, let's analyze the data and the nature of the models:
Here is the provided data over time (in months vs amount):
| Time (months) | Amount owed ($) |
|---------------|-----------------|
| 12 | 2,110 |
| 18 | 1,500 |
| 24 | 870 |
| 30 | 220 |
Step 1: Understand Linear vs Exponential Models:
- Linear Model: In a linear model, the amount decreases by a fixed amount each period. Mathematically, it's represented as [tex]\( Y = aX + b \)[/tex], where [tex]\( Y \)[/tex] is the amount owed, [tex]\( X \)[/tex] is the time, [tex]\( a \)[/tex] is the rate of change, and [tex]\( b \)[/tex] is the initial value.
- Exponential Model: In an exponential model, the amount decreases by a certain percentage over each period. Mathematically, it's represented as [tex]\( Y = b \cdot e^{(aX)} \)[/tex] or [tex]\( Y = b \cdot (1 + r)^X \)[/tex], where [tex]\( Y \)[/tex] is the amount owed, [tex]\( X \)[/tex] is the time, [tex]\( b \)[/tex] is the initial value, [tex]\( e \)[/tex] is Euler's number, [tex]\( a \)[/tex] is the continuous decay rate, and [tex]\( r \)[/tex] is the discrete decay rate.
Step 2: Identify Characteristics of the Decrease:
- From [tex]\( 12 \)[/tex] to [tex]\( 18 \)[/tex] months, the amount owed drops from [tex]\( 2,110 \)[/tex] to [tex]\( 1,500 \)[/tex] — a decrease of [tex]\( 610 \)[/tex].
- From [tex]\( 18 \)[/tex] to [tex]\( 24 \)[/tex] months, it goes from [tex]\( 1,500 \)[/tex] to [tex]\( 870 \)[/tex] — a decrease of [tex]\( 630 \)[/tex].
- From [tex]\( 24 \)[/tex] to [tex]\( 30 \)[/tex] months, it drops from [tex]\( 870 \)[/tex] to [tex]\( 220 \)[/tex] — a decrease of [tex]\( 650 \)[/tex].
This consistent percentage-based decrease strongly suggests an exponential pattern rather than a uniform decrease in dollar amounts, which you'd expect from a linear model.
Step 3: Evaluate Given Statements:
1. The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
- This is not true because the actual decrease is not by an identical amount but by a varying (percentagewise consistent) amount.
2. The linear model better represents the situation because according to the exponential model, the repayment amount will never be 0.
- This isn't a deciding factor for choosing the model. Practically, all debts will be paid off eventually, even in an exponential decay model.
3. The exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months.
- This is incorrect. The decrease is not the same amount each period; it's decreasing by a consistent rate.
4. The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
- This is the most accurate statement. If we project a linear model forward, at some point the model predicts negative amounts, which isn’t realistic for debt repayment. Additionally, the observed rapid decrease fits an exponential model better.
Therefore, the correct statement is: The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
Here is the provided data over time (in months vs amount):
| Time (months) | Amount owed ($) |
|---------------|-----------------|
| 12 | 2,110 |
| 18 | 1,500 |
| 24 | 870 |
| 30 | 220 |
Step 1: Understand Linear vs Exponential Models:
- Linear Model: In a linear model, the amount decreases by a fixed amount each period. Mathematically, it's represented as [tex]\( Y = aX + b \)[/tex], where [tex]\( Y \)[/tex] is the amount owed, [tex]\( X \)[/tex] is the time, [tex]\( a \)[/tex] is the rate of change, and [tex]\( b \)[/tex] is the initial value.
- Exponential Model: In an exponential model, the amount decreases by a certain percentage over each period. Mathematically, it's represented as [tex]\( Y = b \cdot e^{(aX)} \)[/tex] or [tex]\( Y = b \cdot (1 + r)^X \)[/tex], where [tex]\( Y \)[/tex] is the amount owed, [tex]\( X \)[/tex] is the time, [tex]\( b \)[/tex] is the initial value, [tex]\( e \)[/tex] is Euler's number, [tex]\( a \)[/tex] is the continuous decay rate, and [tex]\( r \)[/tex] is the discrete decay rate.
Step 2: Identify Characteristics of the Decrease:
- From [tex]\( 12 \)[/tex] to [tex]\( 18 \)[/tex] months, the amount owed drops from [tex]\( 2,110 \)[/tex] to [tex]\( 1,500 \)[/tex] — a decrease of [tex]\( 610 \)[/tex].
- From [tex]\( 18 \)[/tex] to [tex]\( 24 \)[/tex] months, it goes from [tex]\( 1,500 \)[/tex] to [tex]\( 870 \)[/tex] — a decrease of [tex]\( 630 \)[/tex].
- From [tex]\( 24 \)[/tex] to [tex]\( 30 \)[/tex] months, it drops from [tex]\( 870 \)[/tex] to [tex]\( 220 \)[/tex] — a decrease of [tex]\( 650 \)[/tex].
This consistent percentage-based decrease strongly suggests an exponential pattern rather than a uniform decrease in dollar amounts, which you'd expect from a linear model.
Step 3: Evaluate Given Statements:
1. The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
- This is not true because the actual decrease is not by an identical amount but by a varying (percentagewise consistent) amount.
2. The linear model better represents the situation because according to the exponential model, the repayment amount will never be 0.
- This isn't a deciding factor for choosing the model. Practically, all debts will be paid off eventually, even in an exponential decay model.
3. The exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months.
- This is incorrect. The decrease is not the same amount each period; it's decreasing by a consistent rate.
4. The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
- This is the most accurate statement. If we project a linear model forward, at some point the model predicts negative amounts, which isn’t realistic for debt repayment. Additionally, the observed rapid decrease fits an exponential model better.
Therefore, the correct statement is: The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
