The amount of money in an investment account can be modeled by the function Q, left parenthesis, t, right parenthesis, equals, 7300, left parenthesis, 1, point, 1, right parenthesis, start superscript, 2, t, plus, 2, end superscript, .Q(t)=7300(1.1)

Question

2t+2
. Write an equivalent function of the form Q, left parenthesis, t, right parenthesis, equals, a, b, to the power t , .Q(t)=ab
t
.
Answer
Attempt 2 out of 2
Q, left parenthesis, t, right parenthesis, equalsQ(t)=
left parenthesis(
right parenthesis, to the power t)
t

Answer 1

To express the function \( Q(t) = 7300(1.1)^{2t + 2} \) in the form \( Q(t) = ab^t \), we can manipulate the original equation.

1. First, factor out the constant term: \[ Q(t) = 7300(1.1)^{2t + 2} \]

2. We can rewrite \( (1.1)^{2t + 2} \) as: \[ (1.1)^{2t + 2} = (1.1)^{2t} \cdot (1.1)^{2} = (1.1^2)^{t} \cdot (1.1^{2}) \]

3. Now, calculate \( 1.1^{2} \): \[ 1.1^{2} = 1.21 \]

4. Therefore, we can now write: \[ Q(t) = 7300 \cdot 1.21^{t} \cdot 1.1^{2} \]

5. Which simplifies to: \[ Q(t) = 7300 \times 1.21 \times (1.1^2)^{t} \] Here, you have \( a = 7300 \times 1.21 \) and \( b = 1.1^{2} \).

6. Finally, calculate \( a \): \[ a = 7300 \times 1.21 = 8833 \]

So now we have: \[ Q(t) = 8833(1.1^2)^{t} = 8833(1.21)^{t} \]

Thus, the equivalent function in the required form is:

\[ Q(t) = 8833(1.21)^{t} \]