The quantity of an element is decaying over time such that the amount of material at any time is given by the function B, left parenthesis, t, right parenthesis, equals, 400, left parenthesis, 0, point, 9, right parenthesis, start superscript, 2, t, plus, 2, end superscript, .B(t)=400(0.9)

Question

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

Answer 1

To rewrite the function \( B(t) = 400(0.9)^{2t + 2} \) in the form \( B(t) = ab^t \), we can take the following steps:

1. Factor out constants: Notice that \( 2t + 2 \) can be rewritten as: \[ 2t + 2 = 2(t + 1) \] Therefore, we can rewrite the function as: \[ B(t) = 400(0.9)^{2(t + 1)} = 400(0.9^2)^{t + 1} \]

2. Calculate \( 0.9^2 \): Now calculate \( 0.9^2 \): \[ 0.9^2 = 0.81 \] Hence, we can substitute back into the equation: \[ B(t) = 400(0.81)^{t + 1} \]

3. Express in terms of \( b^t \): Split \( (0.81)^{t + 1} \) into two parts: \[ B(t) = 400 \cdot 0.81^1 \cdot 0.81^t = 400 \cdot 0.81 \cdot (0.81^t) \]

4. Set \( a \) and \( b \): Now we can define: \[ a = 400 \cdot 0.81 = 324 \quad \text{and} \quad b = 0.81 \]

Thus, the function in the form \( B(t) = ab^t \) is:

\[ B(t) = 324(0.81)^t \]