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 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 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.

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 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.
Skip to navigation

Answer 1

To analyze Callie's investment of $1,000 at a 9% interest rate compounded annually, we can look at the formula provided:

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

This is an exponential growth function where \( a(t) \) is the amount in the account after \( t \) years.

Let's determine what happens after 10 years:

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

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

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

Now, let's calculate the amount:

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

After 10 years, the amount in the account will be approximately $2,367.36.

Now, we evaluate which graph fits this situation.

1. The graph must show an increasing curve because the amount in the account grows over time.
2. The amount at \( t = 10 \) years must be around $2,367, which means we need to look for a graph where the y-coordinate at \( x = 10 \) is close to but below $2,500.
3. The x-axis ranges must accommodate at least up to 10 years.

Let's analyze the options:

• The first graph has the x-axis ranging from 0 to 10 and the y-axis from 0 to 2,000. The maximum height is too low as it does not reach 2,367.
• The second graph looks to range from 0 to 14 on the x-axis and 0 to 3,500 on the y-axis, which likely allows for the required values since it can cover up to 2,367.
• The third graph shows a decreasing curve, which is not applicable here as the amount is increasing.
• The fourth graph has the x-axis from 0 to 5, which is not sufficient to cover 10 years.

From this analysis, the second graph, which depicts an increasing curve covering a more extensive time scale and giving a higher range on the y-axis, is the accurate representation of Callie's investment situation.