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 \( Q(t) = ab^t \): \[ a = 8823, , b = 1.21 \]
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