Question
Context/Problem Description:
Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5000⋅1.06
x
. With Option #2, his money will grow accordingly to the function q(x)=500x+5000. He creates sequences for both functions.
Table Data:
Term Number 1 2 3 4 5 6 7 8 9 10
Sequence for Option #1 5,300 5,618 5,955.1 6,312.4 6,691.1 7,092.6 7,518.2 7,969.3 8,447.4 8,954.2
Sequence for Option #2 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 10,000
Export to Sheets
Questions:
Which option has the greater rate of change between years 7 and 8? Option # ____
Which option should he avoid so he can have more money? Option # __
Miguel is planning to invest $5,000. He plans to withdraw his investment and earnings in approximately 10 years. With Option #1, his money will grow according to the function p(x)=5000⋅1.06
x
. With Option #2, his money will grow accordingly to the function q(x)=500x+5000. He creates sequences for both functions.
Table Data:
Term Number 1 2 3 4 5 6 7 8 9 10
Sequence for Option #1 5,300 5,618 5,955.1 6,312.4 6,691.1 7,092.6 7,518.2 7,969.3 8,447.4 8,954.2
Sequence for Option #2 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 10,000
Export to Sheets
Questions:
Which option has the greater rate of change between years 7 and 8? Option # ____
Which option should he avoid so he can have more money? Option # __
Answers
GPT-4o mini
To determine which option has the greater rate of change between years 7 and 8, we can calculate the rate of change for both options.
### For Option #1 (p(x)):
- Value at year 7: \( p(7) = 7,518.2 \)
- Value at year 8: \( p(8) = 7,969.3 \)
Rate of change for Option #1:
\[
\text{Rate of change} = p(8) - p(7) = 7,969.3 - 7,518.2 = 451.1
\]
### For Option #2 (q(x)):
- Value at year 7: \( q(7) = 8,500 \)
- Value at year 8: \( q(8) = 9,000 \)
Rate of change for Option #2:
\[
\text{Rate of change} = q(8) - q(7) = 9,000 - 8,500 = 500
\]
### Comparing the Rates of Change:
- Option #1: 451.1
- Option #2: 500
Since 500 (Option #2) is greater than 451.1 (Option #1), the greater rate of change between years 7 and 8 is for **Option #2**.
### Conclusion for the first question:
**Which option has the greater rate of change between years 7 and 8?**
- Option # **2**
### Second Question:
To determine which option Miguel should avoid for maximum earnings in 10 years, we simply compare the value of the investments at year 10.
- Value for Option #1 at year 10: \( p(10) = 8,954.2 \)
- Value for Option #2 at year 10: \( q(10) = 10,000 \)
Since \( q(10) = 10,000 \) is greater than \( p(10) = 8,954.2 \), Miguel should avoid **Option #1**.
### Conclusion for the second question:
**Which option should he avoid so he can have more money?**
- Option # **1**
### For Option #1 (p(x)):
- Value at year 7: \( p(7) = 7,518.2 \)
- Value at year 8: \( p(8) = 7,969.3 \)
Rate of change for Option #1:
\[
\text{Rate of change} = p(8) - p(7) = 7,969.3 - 7,518.2 = 451.1
\]
### For Option #2 (q(x)):
- Value at year 7: \( q(7) = 8,500 \)
- Value at year 8: \( q(8) = 9,000 \)
Rate of change for Option #2:
\[
\text{Rate of change} = q(8) - q(7) = 9,000 - 8,500 = 500
\]
### Comparing the Rates of Change:
- Option #1: 451.1
- Option #2: 500
Since 500 (Option #2) is greater than 451.1 (Option #1), the greater rate of change between years 7 and 8 is for **Option #2**.
### Conclusion for the first question:
**Which option has the greater rate of change between years 7 and 8?**
- Option # **2**
### Second Question:
To determine which option Miguel should avoid for maximum earnings in 10 years, we simply compare the value of the investments at year 10.
- Value for Option #1 at year 10: \( p(10) = 8,954.2 \)
- Value for Option #2 at year 10: \( q(10) = 10,000 \)
Since \( q(10) = 10,000 \) is greater than \( p(10) = 8,954.2 \), Miguel should avoid **Option #1**.
### Conclusion for the second question:
**Which option should he avoid so he can have more money?**
- Option # **1**