To determine which graph accurately displays the situation, we can evaluate how the amount in the account changes over time according to the equation provided:
\[ a(t) = 1,000(1.09)^t \]
This equation indicates that the amount in the account, \(a(t)\), increases exponentially over time due to the 9% interest rate compounded annually. Since we are interested in the amount after 10 years, we can substitute \(t = 10\):
\[ a(10) = 1,000(1.09)^{10} \]
Calculating \(1.09^{10}\):
\[ 1.09^{10} \approx 2.36736 \quad (\text{approximately}) \]
Now, calculating \(a(10)\):
\[ a(10) \approx 1,000 \times 2.36736 \approx 2367.36 \]
At \(t = 10\), Callie will have approximately $2,367.36 in her account.
Now we need to analyze the provided graph options. We are looking for:
- An increasing curve (indicating that the amount grows over time).
- A range for \(t\) that includes up to 10 years, which means the x-axis should be appropriate (at least up to 10).
- A y-axis range that can accommodate at least $2,367.36, hence extending up to at least $2,500 or $3,000.
Based on the descriptions of the graphs:
- The graph defined by the x-axis ranging from 0 to 14 and y-axis from 0 to 3,500 is an increasing curve.
- The graph defined by the x-axis ranging from 0 to 12 and y-axis from 0 to 1,200 is a decreasing curve, which is incorrect.
- The graph defined by the x-axis ranging from 0 to 5 and y-axis from 0 to 2,000 is an increasing curve, but it doesn't reach enough on the y-axis.
- The graph defined by the x-axis ranging from 0 to 10 and y-axis from 0 to 2,000 is also an increasing curve but won't show values above 2,000.
The first graph is the only one that accurately reflects the increasing amount over the decades and extends high enough into the $3,500 range to encompass the amount Callie will have after 10 years. Thus:
The correct graph is the one with an increasing curve in the first quadrant with the x-axis ranging from 0 to 14 in increments of 2 and the y-axis ranging from 0 to 3,500 in increments of 500.
1 answer