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

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 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.

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.

GPT-4o mini · Answer

To determine which graph accurately represents the situation with Callie’s investment, we first need to evaluate the amount in her account after 10 years.

The formula given for the amount in the account after $ t $ years is:

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

If we want to find the amount after 10 years:

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

Calculating $ (1.09)^{10} $:

$ (1.09)^{10} \approx 2.36736 $ (using a calculator).
So,

\[ a(10) \approx 1,000 \times 2.36736 \approx 2,367.36 \]

This tells us that, after 10 years, the amount in the account will be approximately $2,367.36.

Next, we need to find which graph reflects this information. Here are the considerations for each graph option based on the expected outcome:

First Graph: An increasing curve with y-values reaching up to 2,000. This is unlikely since at 10 years it will exceed 2,000.
Second Graph: A decreasing curve. This is not correct as the investment is growing, not decreasing.
Third Graph: An increasing curve with y-values reaching up to 3,500. This could be correct since $2,367.36 falls within this range, and the curve is increasing.
Fourth Graph: An increasing curve but y-values only go up to 2,000, similar to the first graph. This is also unlikely for the same reason.

Based on the calculations and the available options, the third graph (an increasing curve with y-axis reaching up to 3,500) accurately represents the situation.