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Answer :
The function provided is [tex]\( f(x) = 1,600(1.35)^x \)[/tex], which describes the number of cells in a culture after [tex]\( x \)[/tex] hours.
Let's analyze the function to understand each part:
1. Initial Number of Cells: In an exponential growth function of the form [tex]\( f(x) = a(b)^x \)[/tex], the initial amount is represented by the coefficient [tex]\( a \)[/tex]. In this case, [tex]\( a = 1,600 \)[/tex]. This means that the initial number of cells in the culture is 1,600.
2. Growth Rate: The base of the exponent, [tex]\( b = 1.35 \)[/tex], represents the growth factor per hour. To find the growth rate as a percentage, we subtract 1 from the growth factor and then convert it to a percentage:
- Growth Rate = [tex]\((1.35 - 1) \times 100\% = 0.35 \times 100\% = 35\%\)[/tex].
This indicates that the number of cells increases by 35% per hour.
With these observations:
- Option (A) states that the number of cells increases at a rate of 135% per hour, which is not correct according to our calculation.
- Option (B) states the number of cells increases by 135 per hour, which is incorrect as it misunderstands the concept of exponential growth.
- Option (C) claims that the initial number of cells is 2,160, which does not match our initial value of 1,600.
- Option (D) accurately describes the initial number of cells as 1,600.
Therefore, the correct statement is:
(D) The initial number of cells in the culture is 1,600.
Let's analyze the function to understand each part:
1. Initial Number of Cells: In an exponential growth function of the form [tex]\( f(x) = a(b)^x \)[/tex], the initial amount is represented by the coefficient [tex]\( a \)[/tex]. In this case, [tex]\( a = 1,600 \)[/tex]. This means that the initial number of cells in the culture is 1,600.
2. Growth Rate: The base of the exponent, [tex]\( b = 1.35 \)[/tex], represents the growth factor per hour. To find the growth rate as a percentage, we subtract 1 from the growth factor and then convert it to a percentage:
- Growth Rate = [tex]\((1.35 - 1) \times 100\% = 0.35 \times 100\% = 35\%\)[/tex].
This indicates that the number of cells increases by 35% per hour.
With these observations:
- Option (A) states that the number of cells increases at a rate of 135% per hour, which is not correct according to our calculation.
- Option (B) states the number of cells increases by 135 per hour, which is incorrect as it misunderstands the concept of exponential growth.
- Option (C) claims that the initial number of cells is 2,160, which does not match our initial value of 1,600.
- Option (D) accurately describes the initial number of cells as 1,600.
Therefore, the correct statement is:
(D) The initial number of cells in the culture is 1,600.
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