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 graoh accurately displays the situation.
Based on the given equation a(t) = 1,000(1.09)^t, let's plug in some values for t to see how the amount in the account changes over time:
When t = 0 (immediately after winning the scholarship), a(0) = 1,000(1.09)^0 = 1,000(1) = 1,000.
After 1 year, a(1) = 1,000(1.09)^1 ≈ 1,090.
After 2 years, a(2) = 1,000(1.09)^2 ≈ 1,188.10.
After 3 years, a(3) = 1,000(1.09)^3 ≈ 1,295.03.
And so on...
From these calculations, we can conclude that as t increases, the amount in the account also increases. Therefore, the graph of a(t) should be an increasing curve.
Now let's analyze the given graph options:
Option A: The graph is a straight line. This is incorrect because a(t) should increase as t increases, not remain constant.
Option B: The graph is a decreasing curve. This is incorrect because a(t) should increase as t increases, not decrease.
Option C: The graph is rapidly increasing. This is incorrect because a(t) should increase gradually, but this graph shows a steep increase.
Option D: The graph starts at (0, 1,000), which corresponds to the initial amount in the account, and then increases gradually as t increases. This is correct based on the equation and our analysis above.
Therefore, the correct graph would be Option D.