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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

, 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 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.
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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1 answer

To determine which graph accurately displays the situation based on the equation \( a(t) = 1,000(1.09)^t \), we need to analyze the behavior of the function over time.

  1. Understanding the function: This is an exponential growth function because \( 1.09 \) (the growth factor) is greater than 1. The amount in the account, \( a(t) \), increases as \( t \) (the number of years) increases.

  2. Calculating values: To have a rough idea of how the function behaves over time, let's calculate \( a(t) \) for \( t = 0 \) to \( t = 10 \):

    • \( a(0) = 1,000(1.09)^0 = 1,000 \)
    • \( a(1) = 1,000(1.09)^1 = 1,090 \)
    • \( a(2) = 1,000(1.09)^2 \approx 1,188.10 \)
    • \( a(3) = 1,000(1.09)^3 \approx 1,295.03 \)
    • \( a(4) = 1,000(1.09)^4 \approx 1,411.58 \)
    • \( a(5) = 1,000(1.09)^5 \approx 1,538.62 \)
    • \( a(6) = 1,000(1.09)^6 \approx 1,676.74 \)
    • \( a(7) = 1,000(1.09)^7 \approx 1,826.50 \)
    • \( a(8) = 1,000(1.09)^8 \approx 1,988.70 \)
    • \( a(9) = 1,000(1.09)^9 \approx 2,164.12 \)
    • \( a(10) = 1,000(1.09)^{10} \approx 2,354.07 \)

  3. Analyzing the data: From the calculations, we see that after 10 years, Callie will have approximately $2,354.07 in the account. This shows that \( a(t) \) is indeed increasing over time, particularly in the range of \( 0 \) to \( 10 \) years.

  4. Identifying the correct graph: Given the options provided:

    • The first graph's y-axis goes up to 3,500 and the x-axis up to 14, which can show growth beyond 10 years, but it's not specific to the relevant range.
    • The second graph only covers up to 5 years, which is insufficient for our needs.
    • The third graph covers from 0 to 10 years on the x-axis, with a y-axis that goes up to 2,000. This is relevant because \( a(10) \approx 2,354.07 \). However, the y-axis limit is too low; it should ideally extend beyond 2,000.
    • The fourth graph shows a decreasing curve, which is not applicable for this situation.

In summary, while none of the graphs perfectly represent the data due to potential shortfalls in visual scale, the third graph is the most appropriate choice because it covers the relevant time period (up to 10 years) and indicates increasing values. However, the y-axis scale should ideally be adjusted to reflect values beyond 2,000 to match the output.