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

, 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

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: 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.

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: 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.

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 5 in increments of 1 and the y-axis ranges from 0 to 2000 in increments of 500.
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 5 in increments of 1 and the y-axis ranges from 0 to 2000 in increments of 500.

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 10 in increments of 1 and the y-axis ranges from 0 to 2,000 in increments of 500.

Answer 1

The scenario describes an investment in an account earning 9% interest compounded annually, with a formula given by:

\[ a(t) = 1000(1.09)^t \]

To determine the correct graph displaying Callie's scholarship amount after varying years, we need to keep the following points in mind:

1. The function \( a(t) \) represents exponential growth, since the base (1.09) is greater than 1.
2. We are interested in the time period where \( t \) ranges from 0 to 10 years, as Callie plans to use the scholarship after 10 years.
3. We also note that the expected values for \( a(t) \) should increase over time, reaching a maximum at \( t = 10 \).

Let's evaluate \( a(10) \):

\[ a(10) = 1000(1.09)^{10} \]

Calculating \( (1.09)^{10} \):

\[ (1.09)^{10} \approx 2.367 \]

Thus,

\[ a(10) \approx 1000 \times 2.367 \approx 2367 \]

Now, looking at the possible graphs:

• We know \( a(t) \) will represent an increasing curve for \( t = 0 \) to \( t = 10 \).
• The y-axis should scale to accommodate values around 2367 at \( t = 10 \), which means the graph should extend closer to this value.

Now let's analyze the options:

1. The first option shows a decreasing curve, which is incorrect.
2. The second option shows an increasing curve that scales well to over 3,500, making it plausible for our values.
3. The third option is limited to a smaller range (0 to 5) and won't demonstrate the increase well, as it won't reach \( t=10 \).
4. The fourth option shows an increasing curve with a y-axis up to 2000, which won't contain our value of roughly 2367.

Given these analyses, the correct graph that accurately displays the situation with the expected increasing trend and proper value scale for \( t = 10 \) is the second graph.