Answer :
Sure! Let's go through the solution for this question step-by-step.
1. Identify the Type of Relationship:
- Situation Description: The lake initially has 4 square feet of bacteria, and this area increases by a factor of 3 every hour.
- Analysis: Since the bacteria area grows by a constant factor (3) every hour, this is an example of exponential growth. In exponential growth, the quantity increases by a consistent percentage (or factor) over equal increments of time.
2. Determine the Domain of the Function:
- The domain of a function includes all possible values of the independent variable, which in this case is time (t), measured in hours.
- Since time cannot be negative, the domain consists of all non-negative real numbers. Therefore, the domain is [tex]\( t \geq 0 \)[/tex].
3. Write the Explicit Function:
- Initial Area of Bacteria (A\_0): 4 square feet.
- Growth Factor: 3 (area triples every hour).
- The function that models the relationship between time in hours (t) and the area of bacteria (A) is given by:
[tex]\[
A(t) = 4 \times 3^t
\][/tex]
- This function helps calculate the area covered by bacteria after t hours.
In summary, the situation is exponential because the bacteria area grows by a constant factor every hour. The domain of the function is [tex]\( t \geq 0 \)[/tex], and the explicit function modeling this growth is [tex]\( A(t) = 4 \times 3^t \)[/tex].
1. Identify the Type of Relationship:
- Situation Description: The lake initially has 4 square feet of bacteria, and this area increases by a factor of 3 every hour.
- Analysis: Since the bacteria area grows by a constant factor (3) every hour, this is an example of exponential growth. In exponential growth, the quantity increases by a consistent percentage (or factor) over equal increments of time.
2. Determine the Domain of the Function:
- The domain of a function includes all possible values of the independent variable, which in this case is time (t), measured in hours.
- Since time cannot be negative, the domain consists of all non-negative real numbers. Therefore, the domain is [tex]\( t \geq 0 \)[/tex].
3. Write the Explicit Function:
- Initial Area of Bacteria (A\_0): 4 square feet.
- Growth Factor: 3 (area triples every hour).
- The function that models the relationship between time in hours (t) and the area of bacteria (A) is given by:
[tex]\[
A(t) = 4 \times 3^t
\][/tex]
- This function helps calculate the area covered by bacteria after t hours.
In summary, the situation is exponential because the bacteria area grows by a constant factor every hour. The domain of the function is [tex]\( t \geq 0 \)[/tex], and the explicit function modeling this growth is [tex]\( A(t) = 4 \times 3^t \)[/tex].
