The quantity of an element is decaying over time such that the amount of material at any time is given by the function R, left parenthesis, t, right parenthesis, equals, 4000, left parenthesis, 0, point, 9, right parenthesis, start superscript, t, plus, 1, end superscript, .R(t)=4000(0.9)

Question

t+1
. Write an equivalent function of the form R, left parenthesis, t, right parenthesis, equals, a, b, to the power t , .R(t)=ab
t
.

Answer 1

To rewrite the function \( R(t) = 4000 (0.9)^{t+1} \) in the form \( R(t) = a b^t \), we start by manipulating the existing function.

1. First, we observe that \( (0.9)^{t+1} \) can be separated into two parts: \[ (0.9)^{t+1} = (0.9)^t \cdot (0.9)^1 = (0.9)^t \cdot 0.9 \]

2. Now we can substitute this back into the original function: \[ R(t) = 4000 \cdot (0.9)^t \cdot 0.9 \]

3. Next, we can simplify this: \[ R(t) = 4000 \cdot 0.9 \cdot (0.9)^t = 3600 \cdot (0.9)^t \]

From this manipulation, we can identify \( a \) and \( b \):

• \( a = 3600 \)
• \( b = 0.9 \)

Thus, the function can be rewritten in the desired form as: \[ R(t) = 3600 (0.9)^t \]