To rewrite the function \( Z(t) = 4000(1.2)^{2t+2} \) in the form \( Z(t) = ab^t \), we need to manipulate the expression.
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Start with the original function: \[ Z(t) = 4000(1.2)^{2t + 2} \]
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We can separate the exponent: \[ Z(t) = 4000(1.2)^{2t} \cdot (1.2)^2 \]
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Calculate \( (1.2)^2 \): \[ (1.2)^2 = 1.44 \]
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Now rewrite the function: \[ Z(t) = 4000 \cdot 1.44 \cdot (1.2)^{2t} \]
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Multiply \( 4000 \) and \( 1.44 \): \[ 4000 \cdot 1.44 = 5760 \]
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Rewrite \( (1.2)^{2t} \) as \( ((1.2)^2)^t \): \[ (1.2)^{2t} = (1.44)^t \]
Putting it all together, we have: \[ Z(t) = 5760 \cdot (1.44)^{t} \]
Thus, in the form \( Z(t) = ab^t \), we have:
- \( a = 5760 \)
- \( b = 1.44 \)
Therefore, the equivalent function is: \[ Z(t) = 5760 \cdot (1.44)^{t} \]