The number of bacteria in a lab experiment can be modeled by the function P, left parenthesis, t, right parenthesis, equals, 220, left parenthesis, 3, right parenthesis, start superscript, t, plus, 2, end superscript, .P(t)=220(3)

Question

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

Answer 1

To rewrite the function \( P(t) = 220(3)^{t+2} \) in the desired form \( P(t) = ab^t \), we can manipulate the existing formula.

First, we can expand the function as follows:

\[ P(t) = 220 \cdot 3^{t+2} \]

Using the properties of exponents, we can rewrite \( 3^{t+2} \) as \( 3^t \cdot 3^2 \):

\[ P(t) = 220 \cdot 3^t \cdot 3^2 \]

Since \( 3^2 = 9 \), we can simplify this to:

\[ P(t) = 220 \cdot 9 \cdot 3^t \]

Now we can calculate \( 220 \cdot 9 \):

\[ 220 \cdot 9 = 1980 \]

Now we can write the function in the standard form \( P(t) = ab^t \):

\[ P(t) = 1980 \cdot 3^t \]

So, the equivalent function is:

\[ P(t) = 1980 \cdot 3^t \]

Here, \( a = 1980 \) and \( b = 3 \).