Question
Use the table to answer the question.
0 375,000
1
2
3
4
5
The Diaz family purchased a home for $375,000. They decide to sell their home five years later. If the value of the home is increasing at a rate of 6% each year, the situation can be modeled using the equation [Math Processing Error] Finish the table to determine which graph displays the correct plotted points for the situation.(1 point)
Responses
A) A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 200,000 to 500,000 in increments of 25,000.
- image with description: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 200,000 to 500,000 in increments of 25,000. - - no response given
B) A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 0 to 350,000 in increments of 50,000.
- image with description: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 0 to 350,000 in increments of 50,000. - - incorrect
C) An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 350,000 to 550,000 in increments of 25,000.
- image with description: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 350,000 to 550,000 in increments of 25,000.
D) An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.
- image with description: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.
0 375,000
1
2
3
4
5
The Diaz family purchased a home for $375,000. They decide to sell their home five years later. If the value of the home is increasing at a rate of 6% each year, the situation can be modeled using the equation [Math Processing Error] Finish the table to determine which graph displays the correct plotted points for the situation.(1 point)
Responses
A) A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 200,000 to 500,000 in increments of 25,000.
- image with description: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 200,000 to 500,000 in increments of 25,000. - - no response given
B) A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 0 to 350,000 in increments of 50,000.
- image with description: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 0 to 350,000 in increments of 50,000. - - incorrect
C) An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 350,000 to 550,000 in increments of 25,000.
- image with description: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for years ranges from 0 to 7 in increments of 1. The y-axis for home value ranges from 350,000 to 550,000 in increments of 25,000.
D) An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.
- image with description: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis is labeled Years and ranges from 350,000 to 550,000 in increments of 25,000. The y-axis is labeled Home Value and ranges from 0 to 7 in increments of 1.
Answers
Answered by
GPT-4o mini
To model the situation where the Diaz family purchased a home for $375,000 and the value increases at a rate of 6% each year, we can use the formula for exponential growth:
\[ V(t) = V_0 \times (1 + r)^t \]
Where:
- \( V(t) \) is the value of the home after \( t \) years,
- \( V_0 \) is the initial value of the home (which is $375,000),
- \( r \) is the growth rate (6% = 0.06),
- \( t \) is the number of years since purchase.
Let's plug in the values to find the home value for years 0 to 5:
1. **Year 0**:
\[ V(0) = 375,000 \times (1 + 0.06)^0 = 375,000 \]
2. **Year 1**:
\[ V(1) = 375,000 \times (1 + 0.06)^1 = 375,000 \times 1.06 = 397,500 \]
3. **Year 2**:
\[ V(2) = 375,000 \times (1 + 0.06)^2 = 375,000 \times 1.1236 \approx 421,350 \]
4. **Year 3**:
\[ V(3) = 375,000 \times (1 + 0.06)^3 = 375,000 \times 1.191016 \approx 446,310 \]
5. **Year 4**:
\[ V(4) = 375,000 \times (1 + 0.06)^4 = 375,000 \times 1.262477 \approx 472,550 \]
6. **Year 5**:
\[ V(5) = 375,000 \times (1 + 0.06)^5 = 375,000 \times 1.338225 \approx 500,000 \]
Now, let's summarize the values in a table format:
| Year (t) | Home Value (V(t)) |
|----------|---------------------|
| 0 | 375,000 |
| 1 | 397,500 |
| 2 | 421,350 |
| 3 | 446,310 |
| 4 | 472,550 |
| 5 | 500,000 |
Now, looking at the options:
- **Option A**: A decreasing curve (incorrect since the value is increasing).
- **Option B**: A decreasing curve (incorrect since the value is increasing).
- **Option C**: An increasing curve (correct since the values are increasing and within the correct range for years 0 to 5).
- **Option D**: Incorrect axes labels (Years on the x-axis and Home Value on the y-axis should be reversed).
Therefore, the correct graph that represents the situation is **Option C**.
\[ V(t) = V_0 \times (1 + r)^t \]
Where:
- \( V(t) \) is the value of the home after \( t \) years,
- \( V_0 \) is the initial value of the home (which is $375,000),
- \( r \) is the growth rate (6% = 0.06),
- \( t \) is the number of years since purchase.
Let's plug in the values to find the home value for years 0 to 5:
1. **Year 0**:
\[ V(0) = 375,000 \times (1 + 0.06)^0 = 375,000 \]
2. **Year 1**:
\[ V(1) = 375,000 \times (1 + 0.06)^1 = 375,000 \times 1.06 = 397,500 \]
3. **Year 2**:
\[ V(2) = 375,000 \times (1 + 0.06)^2 = 375,000 \times 1.1236 \approx 421,350 \]
4. **Year 3**:
\[ V(3) = 375,000 \times (1 + 0.06)^3 = 375,000 \times 1.191016 \approx 446,310 \]
5. **Year 4**:
\[ V(4) = 375,000 \times (1 + 0.06)^4 = 375,000 \times 1.262477 \approx 472,550 \]
6. **Year 5**:
\[ V(5) = 375,000 \times (1 + 0.06)^5 = 375,000 \times 1.338225 \approx 500,000 \]
Now, let's summarize the values in a table format:
| Year (t) | Home Value (V(t)) |
|----------|---------------------|
| 0 | 375,000 |
| 1 | 397,500 |
| 2 | 421,350 |
| 3 | 446,310 |
| 4 | 472,550 |
| 5 | 500,000 |
Now, looking at the options:
- **Option A**: A decreasing curve (incorrect since the value is increasing).
- **Option B**: A decreasing curve (incorrect since the value is increasing).
- **Option C**: An increasing curve (correct since the values are increasing and within the correct range for years 0 to 5).
- **Option D**: Incorrect axes labels (Years on the x-axis and Home Value on the y-axis should be reversed).
Therefore, the correct graph that represents the situation is **Option C**.
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