allie 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 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.

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.

Answer 1

To determine which graph accurately displays the situation described, we'll need to analyze the function \( a(t) = 1000(1.09)^t \) over the relevant period, which is 10 years for Callie's scholarship.

1. Function Analysis: The amount in the account after \( t \) years is modeled by \( a(t) = 1000(1.09)^t \). This is an exponential growth function.

2. Calculate Amount after 10 Years: We need to find \( a(10) \): \[ a(10) = 1000(1.09)^{10} \]

   Using a calculator: \[ (1.09)^{10} \approx 2.3674 \] Therefore: \[ a(10) \approx 1000 \times 2.3674 \approx 2367.40 \]

3. Behavior of the Function: Since this is an exponential function with a positive growth rate, \( a(t) \) continuously increases as \( t \) increases.

4. Graph Selection: Now we examine the options based on the expected behavior of \( a(t) \):

   • The graph should show an increasing curve because the amount grows over time.
   • At \( t = 10 \), the value should be around 2367.40, thus well within the range of 0 to 3,500 on the y-axis.

Evaluating the Options:

1. First Graph: Increasing curve, x-axis from 0 to 14, y-axis from 0 to 3,500 - This graph is suitable.
2. Second Graph: Increasing curve, x-axis from 0 to 5, y-axis from 0 to 2000 - This graph is insufficient since it doesn't reach up to 10 years and limits the range.
3. Third Graph: Decreasing curve - Not applicable since the function is increasing.
4. Fourth Graph: Increasing curve, x-axis from 0 to 10, y-axis from 0 to 2000 - This might show part of the growth, but it doesn't cover the upper bound around 2367.40 adequately.

Conclusion:

The first graph accurately displays the situation because it shows an increasing curve and accommodates the values we expect for \( a(t) \) after 10 years, ranging correctly to 3,500 on the y-axis.