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)

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
.

1 answer

To rewrite the function \( Q(t) = 7300(1.1)^{2t+2} \) in the form \( Q(t) = ab^t \), we need to manipulate the exponent to get it into the desired form.

Starting with the function you provided:

\[ Q(t) = 7300(1.1)^{2t+2} \]

We can separate the exponent \( 2t + 2 \) as follows:

\[ Q(t) = 7300(1.1)^{2t} \cdot (1.1)^2 \]

Now, we know that \( (1.1)^2 = 1.21 \), so we have:

\[ Q(t) = 7300 \cdot 1.21 \cdot (1.1)^{2t} \]

Next, we can calculate \( 7300 \cdot 1.21 \):

\[ 7300 \cdot 1.21 = 8823 \]

Now we can rewrite the function:

\[ Q(t) = 8823 \cdot (1.1)^{2t} \]

At this stage, we have expressed the function in a form that represents \( b^{2t} \). However, to convert it to the form \( ab^t \), we can let:

\[ b = (1.1)^2 \] and thus replace \( (1.1)^{2t} \) by \( (b^t) \):

\[ b = 1.1^2 = 1.21 \]

Now, write \( Q(t) \) in the final form:

\[ Q(t) = 8823 \cdot (1.21)^t \]

Thus, we have:

\[ Q(t) = 8823 \cdot (1.21)^t \]

In conclusion, expressed in the desired form \( Q(t) = ab^t \): \[ a = 8823, , b = 1.21 \]