Exponential Functions from Situations Quick Check

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Question
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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Answer 1

To determine which graph accurately displays the growth of Callie's scholarship after 10 years, we can evaluate the amount in the account using the given exponential function:

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

We want to find \(a(10)\), which represents the amount in the account after 10 years.

Calculating \(a(10)\):

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

Using a calculator to find \( (1.09)^{10} \):

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

Now, substitute this value back into our equation:

\[ a(10) = 1000 \times 2.36736 \approx 2367.36 \]

Now we can look for a graph that shows an increasing curve and has a value around \(2367.36\) at \(t = 10\) years.

Examining the graph options:

1. The first graph shows the y-axis ranging up to 3,500, which accommodates our calculated value.
2. The second graph only goes up to 2,000; thus it is likely not appropriate since \(2,367.36\) won't fit well.
3. The third graph appears to also go up to 2,000; likewise, we can't select that one either.
4. The fourth graph is a decreasing curve, which is incorrect for this situation.

Based on the computed amount and the graph descriptions, the first graph is the only valid choice, as it accurately captures an increasing trend and has a sufficient y-axis range to include the approximate amount after 10 years.

Final Answer: The first graph.