Callie entered an art contest in second grade and won a $1,000 scholarship. The money was invested in an account paying a 9% interest rate compounded annually. The situation can be modeled by the equation a(t)=1,000(1.09)t

Question

𝑎
(
𝑡
)
=
1
,
000
(
1.09
)
𝑡
, where a is the amount in the account after t years. If Callie uses the scholarship 10 years later, determine which graph accurately displays the situation.(1 point)
Responses

Image with alt text: An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 14 in increments of 2 and the y-axis ranges from 0 to 3,500 in increments of 500.

Image with alt text: A decreasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 12 in increments of 2 and the y-axis ranges from 0 to 1,200 in increments of 200.

Image with alt text: An increasing curve with an arrow at the end is plotted

Answer 1

To solve this problem, we need to understand the growth of the investment over time according to the provided exponential growth model:

\[ a(t) = 1,000(1.09)^t \]

Here, \( a(t) \) represents the amount of money in the account after \( t \) years, with the initial amount of $1,000 and an annual interest rate of 9%.

1. Graph Behavior: The function \( a(t) \) is an exponential growth function because the base (1.09) is greater than 1. This means that the amount will increase over time.

2. 10 Years Later: We are specifically interested in the amount after 10 years: \[ a(10) = 1,000(1.09)^{10} \] Calculating \( (1.09)^{10} \): \[ (1.09)^{10} \approx 2.3679 \quad (\text{using a calculator for precision}) \] Now plugging that back into the equation: \[ a(10) \approx 1,000 \times 2.3679 \approx 2,367.90 \]

3. Choosing the Right Graph:

• The first graph is an increasing curve which is likely to be correct as the function models exponential growth.
• The second graph is a decreasing curve, which does not match our function as \( a(t) \) is not decreasing.
• The third graph is described as another increasing curve, implying it's possible as well (assuming it closely aligns with the function).

Based on our calculations and understanding of the function's growth behavior, the correct graph should be the one that shows an increasing curve as \( t \) progresses, confirming that the amount in the account is indeed growing over time, especially around the 10-year mark.

Given that we don't have the images, I recommend you look for the first or third image that shows an increasing curve. The first graph seems likely to accurately represent the situation if it corresponds to the rise in the amount of money over time.