Question
Kamran invests $1800 into an account. The amount of money in his account grows at a rate of 3.1% each year. Kamran builds a model to represent the amount of money in his account over the next 10 years. Which of the following are key features of the graph?
A. The graph has a y-intercept of (0, 1800)
B. The graph is a straight line
C. The graph shows a decreasing function
D: the graph has a relative maximum
A. The graph has a y-intercept of (0, 1800)
B. The graph is a straight line
C. The graph shows a decreasing function
D: the graph has a relative maximum
Answers
GPT-4o mini
To analyze Kamran's investment, we note that he invests $1800 and the account grows at an annual rate of 3.1%. The amount of money in his account can be modeled using the formula for exponential growth:
\[ A(t) = P(1 + r)^t \]
where:
- \( A(t) \) is the amount of money in the account at time \( t \),
- \( P \) is the initial amount invested ($1800),
- \( r \) is the growth rate (0.031), and
- \( t \) is the number of years.
Now, let's evaluate the options based on this model:
A. **The graph has a y-intercept of (0, 1800)** – True. At \( t = 0 \), the amount in the account is \( A(0) = 1800(1 + 0.031)^0 = 1800 \). So the graph will indeed start at the point (0, 1800).
B. **The graph is a straight line** – False. This is an exponential growth function, which is not a straight line. The graph will curve upward, becoming steeper over time.
C. **The graph shows a decreasing function** – False. The graph represents an increasing function because the money in the account is growing over time due to the positive growth rate.
D. **The graph has a relative maximum** – False. The graph continues to grow without bound as time increases, so there is no point where it reaches a maximum and then decreases.
Based on the analysis, the correct features of the graph are:
- **A** is true.
Thus, the answer is A.
\[ A(t) = P(1 + r)^t \]
where:
- \( A(t) \) is the amount of money in the account at time \( t \),
- \( P \) is the initial amount invested ($1800),
- \( r \) is the growth rate (0.031), and
- \( t \) is the number of years.
Now, let's evaluate the options based on this model:
A. **The graph has a y-intercept of (0, 1800)** – True. At \( t = 0 \), the amount in the account is \( A(0) = 1800(1 + 0.031)^0 = 1800 \). So the graph will indeed start at the point (0, 1800).
B. **The graph is a straight line** – False. This is an exponential growth function, which is not a straight line. The graph will curve upward, becoming steeper over time.
C. **The graph shows a decreasing function** – False. The graph represents an increasing function because the money in the account is growing over time due to the positive growth rate.
D. **The graph has a relative maximum** – False. The graph continues to grow without bound as time increases, so there is no point where it reaches a maximum and then decreases.
Based on the analysis, the correct features of the graph are:
- **A** is true.
Thus, the answer is A.