Question

The amount of money in an investment account can be modeled by the function Z, left parenthesis, t, right parenthesis, equals, 4000, left parenthesis, 1, point, 2, right parenthesis, start superscript, 2, t, plus, 2, end superscript, .Z(t)=4000(1.2)
2t+2
. Write an equivalent function of the form Z, left parenthesis, t, right parenthesis, equals, a, b, to the power t , .Z(t)=ab
t
.

Answer 1

To rewrite the function \( Z(t) = 4000(1.2)^{2t+2} \) in the form \( Z(t) = ab^t \), we need to manipulate the expression.

1. Start with the original function:
\[ Z(t) = 4000(1.2)^{2t + 2} \]

2. We can separate the exponent:
\[ Z(t) = 4000(1.2)^{2t} \cdot (1.2)^2 \]

3. Calculate \( (1.2)^2 \):
\[ (1.2)^2 = 1.44 \]

4. Now rewrite the function:
\[ Z(t) = 4000 \cdot 1.44 \cdot (1.2)^{2t} \]

5. Multiply \( 4000 \) and \( 1.44 \):
\[ 4000 \cdot 1.44 = 5760 \]

6. 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:
\[ Z(t) = 5760 \cdot (1.44)^{t} \]