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Answer :
To solve this problem, we need to analyze the function [tex]\( f(x) = 1,600(1.35)^x \)[/tex], which describes the number of cells in a culture after [tex]\( x \)[/tex] hours. We want to determine which statement best describes one value in this function.
Let's break down the function:
1. Identifying the initial number of cells:
- In the function [tex]\( f(x) = 1,600(1.35)^x \)[/tex], the initial number of cells in the culture is represented by the term [tex]\( 1,600 \)[/tex].
- This value is the coefficient in front of the exponential term. It gives the number of cells when [tex]\( x = 0 \)[/tex].
- When [tex]\( x = 0 \)[/tex], the function simplifies to [tex]\( f(0) = 1,600 \times (1.35)^0 = 1,600 \)[/tex].
- Therefore, the initial number of cells in the culture is 1,600.
2. Understanding the growth factor:
- The base of the exponential function, [tex]\( 1.35 \)[/tex], represents the growth factor per hour.
- This means that each hour, the number of cells is multiplied by [tex]\( 1.35 \)[/tex].
3. Analyzing the rate of increase:
- The expression [tex]\( 1.35 \)[/tex] indicates a growth rate of [tex]\( 35\% \)[/tex] per hour. This is because the growth factor [tex]\( 1.35 \)[/tex] can be split into [tex]\( 1 + 0.35 \)[/tex], where [tex]\( 0.35 \)[/tex] is equivalent to [tex]\( 35\% \)[/tex].
- However, the statement option (A) suggests an increase at a rate of [tex]\( 135\% \)[/tex], which is incorrect since it would imply a growth factor of [tex]\( 2.35 \)[/tex].
Let's evaluate the given options:
- Option (A): The number of cells increases at a rate of [tex]\( 135\% \)[/tex] per hour. This is incorrect due to the explanation above.
- Option (B): The number of cells increases by 135 per hour. This is incorrect because the growth is exponential, not linear.
- Option (C): The initial number of cells in the culture is 2,160. This is incorrect, as we've determined the initial number is 1,600.
- Option (D): The initial number of cells in the culture is 1,600. This is correct because it aligns with our analysis.
Thus, the best statement that describes one value in this function is:
(D) The initial number of cells in the culture is 1,600.
Let's break down the function:
1. Identifying the initial number of cells:
- In the function [tex]\( f(x) = 1,600(1.35)^x \)[/tex], the initial number of cells in the culture is represented by the term [tex]\( 1,600 \)[/tex].
- This value is the coefficient in front of the exponential term. It gives the number of cells when [tex]\( x = 0 \)[/tex].
- When [tex]\( x = 0 \)[/tex], the function simplifies to [tex]\( f(0) = 1,600 \times (1.35)^0 = 1,600 \)[/tex].
- Therefore, the initial number of cells in the culture is 1,600.
2. Understanding the growth factor:
- The base of the exponential function, [tex]\( 1.35 \)[/tex], represents the growth factor per hour.
- This means that each hour, the number of cells is multiplied by [tex]\( 1.35 \)[/tex].
3. Analyzing the rate of increase:
- The expression [tex]\( 1.35 \)[/tex] indicates a growth rate of [tex]\( 35\% \)[/tex] per hour. This is because the growth factor [tex]\( 1.35 \)[/tex] can be split into [tex]\( 1 + 0.35 \)[/tex], where [tex]\( 0.35 \)[/tex] is equivalent to [tex]\( 35\% \)[/tex].
- However, the statement option (A) suggests an increase at a rate of [tex]\( 135\% \)[/tex], which is incorrect since it would imply a growth factor of [tex]\( 2.35 \)[/tex].
Let's evaluate the given options:
- Option (A): The number of cells increases at a rate of [tex]\( 135\% \)[/tex] per hour. This is incorrect due to the explanation above.
- Option (B): The number of cells increases by 135 per hour. This is incorrect because the growth is exponential, not linear.
- Option (C): The initial number of cells in the culture is 2,160. This is incorrect, as we've determined the initial number is 1,600.
- Option (D): The initial number of cells in the culture is 1,600. This is correct because it aligns with our analysis.
Thus, the best statement that describes one value in this function is:
(D) The initial number of cells in the culture is 1,600.
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