Question
page 19 of 19
Comparing & Creating Models Quick Check
2 of 52 of 5 Items
Question
Use the table to answer the question.
Years, x 0 5 10 15
Money in Account, y (in $) 30,000.00 49,461.64 81,548.45 134,450.67
Norma seeks out the help of a financial advisor to discuss plans that will help her earn money over time on an initial investment. The advisor offers her two plans that will allow her to retire at the age of 50. The first investment option can be modeled by the equation y=20,000e0.12x
, while the second investment option is modeled by the table. Compare the two function models. If Norma wants to invest as little money as possible up front, which option should she choose?
(1 point)
Responses
Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $13,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $13,000.00 more than the first option after 10 years.
Norma should select the first investment option since it has a y-intercept of $20,000.00.
Norma should select the first investment option since it has a y -intercept of $20,000.00.
Norma should select the second investment option since it has a y-intercept of $30,000.00.
Comparing & Creating Models Quick Check
2 of 52 of 5 Items
Question
Use the table to answer the question.
Years, x 0 5 10 15
Money in Account, y (in $) 30,000.00 49,461.64 81,548.45 134,450.67
Norma seeks out the help of a financial advisor to discuss plans that will help her earn money over time on an initial investment. The advisor offers her two plans that will allow her to retire at the age of 50. The first investment option can be modeled by the equation y=20,000e0.12x
, while the second investment option is modeled by the table. Compare the two function models. If Norma wants to invest as little money as possible up front, which option should she choose?
(1 point)
Responses
Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $13,000.00 more than the first option after 10 years.
Norma should select the second investment option since it will give her around $13,000.00 more than the first option after 10 years.
Norma should select the first investment option since it has a y-intercept of $20,000.00.
Norma should select the first investment option since it has a y -intercept of $20,000.00.
Norma should select the second investment option since it has a y-intercept of $30,000.00.
Answers
GPT-4o mini
To compare the two investment options, we will evaluate the amount available in each option after 10 years.
### For the first investment option:
The equation for the first investment is:
\[ y = 20,000e^{0.12x} \]
Substituting \( x = 10 \):
\[ y = 20,000e^{0.12 \times 10} \]
\[ = 20,000e^{1.2} \]
Calculating \( e^{1.2} \):
Approximately, \( e^{1.2} \approx 3.320 \).
Now plug this value into the equation:
\[ y \approx 20,000 \times 3.320 \]
\[ y \approx 66,400 \]
### For the second investment option:
From the table, when \( x = 10 \) (years), the money in the account is:
\[ y \approx 81,548.45 \]
### Comparing the two:
- First investment option after 10 years: **$66,400**
- Second investment option after 10 years: **$81,548.45**
### Difference:
To find the difference:
\[ \text{Difference} = 81,548.45 - 66,400 = 15,148.45 \]
Thus, the second investment option gives Norma approximately **$15,148.45** more than the first one after 10 years.
### Decision:
Since Norma wants to invest as little money as possible up front, she should select the **second investment option**, as it offers a higher return after 10 years, despite the higher initial investment of $30,000 compared to the first option's initial investment of $20,000.
The correct response is:
**Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.**
### For the first investment option:
The equation for the first investment is:
\[ y = 20,000e^{0.12x} \]
Substituting \( x = 10 \):
\[ y = 20,000e^{0.12 \times 10} \]
\[ = 20,000e^{1.2} \]
Calculating \( e^{1.2} \):
Approximately, \( e^{1.2} \approx 3.320 \).
Now plug this value into the equation:
\[ y \approx 20,000 \times 3.320 \]
\[ y \approx 66,400 \]
### For the second investment option:
From the table, when \( x = 10 \) (years), the money in the account is:
\[ y \approx 81,548.45 \]
### Comparing the two:
- First investment option after 10 years: **$66,400**
- Second investment option after 10 years: **$81,548.45**
### Difference:
To find the difference:
\[ \text{Difference} = 81,548.45 - 66,400 = 15,148.45 \]
Thus, the second investment option gives Norma approximately **$15,148.45** more than the first one after 10 years.
### Decision:
Since Norma wants to invest as little money as possible up front, she should select the **second investment option**, as it offers a higher return after 10 years, despite the higher initial investment of $30,000 compared to the first option's initial investment of $20,000.
The correct response is:
**Norma should select the second investment option since it will give her around $15,000.00 more than the first option after 10 years.**