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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Answer 1

To determine which graph accurately displays the situation after Callie invests her $1,000 scholarship at a 9% interest rate compounded annually, we can analyze the situation using the formula provided:

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

Here, \( a(t) \) is the amount in the account after \( t \) years. Since Callie uses the scholarship 10 years later, we must evaluate the function at \( t = 10 \):

\[ a(10) = 1,000(1.09)^{10} \]

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

\[ (1.09)^{10} \approx 2.36736 \]

Now plugging it back into the equation:

\[ a(10) \approx 1,000 \times 2.36736 \approx 2367.36 \]

This means that after 10 years, the amount in the account is approximately $2367.36.

Now we analyze the graph options provided:

1. First Graph: X-axis ranges from 0 to 5 and Y-axis from 0 to 2000. This doesn't cover 10 years or amounts over $2000.
2. Second Graph: X-axis from 0 to 14 and Y-axis from 0 to 3,500. This can cover up to 10 years and amounts over $2000.
3. Third Graph: X-axis from 0 to 10 and Y-axis from 0 to 2000. This does not reach the approximately $2367.36 amount.
4. Fourth Graph: A decreasing curve, which does not represent our situation of growing savings with compounded interest.

Given this analysis, the second graph accurately displays the situation, as it covers both a sufficient range for years (up to 14) and amounts (up to 3,500), which includes the approximated value of $2367.36 after 10 years.

The correct response is the second graph.