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

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

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.
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is it a b c or d?

Answer 1

To determine which graph accurately displays Callie's situation with her scholarship, we can analyze the equation given:

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

This indicates that her investment of $1,000 grows over time at an annual interest rate of 9%, compounded yearly.

1. Interest Rate: Since the interest is compounded annually at 9%, the graph will be an increasing exponential curve.

2. X-Axis (Time): The x-axis should represent time in years. Since Callie uses the scholarship 10 years later, we will particularly pay attention to values around 10 years.

3. Y-Axis (Amount): We can calculate the amount after 10 years to see how high the y-value will go: \[ a(10) = 1,000(1.09)^{10} \approx 1,000 \times 2.36736 \approx 2368 \] So, after 10 years the amount will be approximately $2,368.

Given these characteristics, we look at the response options:

• Option A: X ranges from 0 to 5, which does not cover the 10 years needed.
• Option B: X ranges from 0 to 14; possible but we need to consider the y-axis.
• Option C: X ranges from 0 to 10 and the y-axis goes up to 2000, which would not be sufficient as we calculated around $2,368.
• Option D: A decreasing curve is not applicable here as money in an investment account should not decrease over time.

Thus, the correct choice is Option B, which shows an increasing curve extending beyond 10 years and allows for values beyond $2,000, making it the most appropriate graph.