Question
The amount of revenue for a business can be modeled by the function D, left parenthesis, t, right parenthesis, equals, 3500, left parenthesis, 2, right parenthesis, start superscript, t, plus, 1, end superscript, .D(t)=3500(2)
t+1
. Write an equivalent function of the form D, left parenthesis, t, right parenthesis, equals, a, b, to the power t , .D(t)=ab
t
.
t+1
. Write an equivalent function of the form D, left parenthesis, t, right parenthesis, equals, a, b, to the power t , .D(t)=ab
t
.
Answers
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Answered by
GPT-4o mini
To write the function \( D(t) = 3500(2)^{t+1} \) in the form \( D(t) = ab^t \), we need to manipulate the expression a bit.
Starting with:
\[
D(t) = 3500(2)^{t+1}
\]
We can rewrite \( (2)^{t+1} \) as \( (2^t)(2^1) \):
\[
D(t) = 3500 \cdot (2^t) \cdot 2
\]
Now, simplifying it gives us:
\[
D(t) = 3500 \cdot 2 \cdot 2^t
\]
Calculating \( 3500 \cdot 2 \) gives us:
\[
3500 \cdot 2 = 7000
\]
So, we can rewrite the function as:
\[
D(t) = 7000 \cdot 2^t
\]
Thus, in the form \( D(t) = ab^t \), we have:
- \( a = 7000 \)
- \( b = 2 \)
Therefore, the equivalent function is:
\[
D(t) = 7000 \cdot 2^t
\]
Starting with:
\[
D(t) = 3500(2)^{t+1}
\]
We can rewrite \( (2)^{t+1} \) as \( (2^t)(2^1) \):
\[
D(t) = 3500 \cdot (2^t) \cdot 2
\]
Now, simplifying it gives us:
\[
D(t) = 3500 \cdot 2 \cdot 2^t
\]
Calculating \( 3500 \cdot 2 \) gives us:
\[
3500 \cdot 2 = 7000
\]
So, we can rewrite the function as:
\[
D(t) = 7000 \cdot 2^t
\]
Thus, in the form \( D(t) = ab^t \), we have:
- \( a = 7000 \)
- \( b = 2 \)
Therefore, the equivalent function is:
\[
D(t) = 7000 \cdot 2^t
\]
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