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 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.

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.

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.

Answer 1

To determine which graph accurately displays Callie's investment situation, we can analyze the mathematical model given:

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

This equation represents the growth of the investment over time with a 9% annual interest rate compounded annually.

1. Initial Value: At \( t = 0 \) (the time of the contest), the amount is \( a(0) = 1,000(1.09)^0 = 1,000 \).

2. Growth Over Time: After 10 years, we can calculate the amount in the account: \[ a(10) = 1,000(1.09)^{10} \] \[ a(10) \approx 1,000 \times 2.3674 \approx 2,367.40 \]

   So, after 10 years, the value in the account is approximately $2,367.40.

3. Analysis of the Graphs:

   • The first option extends beyond 10 years up to 14, with y-axis values up to 3,500, and shows continuous growth, indicating it could represent the compound interest growth over an extended period.
   • The second option ranges only up to 10 years and values reaching 2,000, which might not accurately reflect the amount calculated (2,367.40).
   • The third option is limited to 5 years, which is too short to show the growth properly.
   • The fourth option shows a decreasing curve, which does not match the compounding growth of the investment at all.

Given these observations, the best choice would be the first option, as it shows an increasing curve with an appropriate y-axis range that could extend well above 2,367.40, allowing for sufficient growth over the 10 years.