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Answer :
To find which function best models the data for the percentage of the house left to build over time, we need to determine a linear model using the given data:
The data points are:
- (0, 100)
- (1, 86)
- (2, 65)
- (3, 59)
- (4, 41)
- (5, 34)
A linear relationship can be represented by the equation of a line: [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
### Step-by-step Breakdown:
1. Identify the Slope (m):
The slope [tex]\( m \)[/tex] tells us how much the percentage of the house left to build decreases each month.
We have the result: [tex]\( m \approx -13.457 \)[/tex]
2. Identify the Y-intercept (c):
The y-intercept [tex]\( c \)[/tex] is the percentage of the house left to build at the start (when [tex]\( x = 0 \)[/tex]).
We have the result: [tex]\( c \approx 97.81 \)[/tex]
3. Match with Given Options:
Now, compare these results to the options given:
- A) [tex]\( f(x) = -13.5x + 97.8 \)[/tex]
- B) [tex]\( f(x) = -13.5x + 7.3 \)[/tex]
- C) [tex]\( f(x) = 97.8x - 13.5 \)[/tex]
- D) [tex]\( f(x) = 7.3x - 97.8 \)[/tex]
The closest match based on both the slope and the y-intercept is option A, [tex]\( f(x) = -13.5x + 97.8 \)[/tex].
Therefore, the best function to model the data is option A: [tex]\( f(x) = -13.5x + 97.8 \)[/tex].
The data points are:
- (0, 100)
- (1, 86)
- (2, 65)
- (3, 59)
- (4, 41)
- (5, 34)
A linear relationship can be represented by the equation of a line: [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
### Step-by-step Breakdown:
1. Identify the Slope (m):
The slope [tex]\( m \)[/tex] tells us how much the percentage of the house left to build decreases each month.
We have the result: [tex]\( m \approx -13.457 \)[/tex]
2. Identify the Y-intercept (c):
The y-intercept [tex]\( c \)[/tex] is the percentage of the house left to build at the start (when [tex]\( x = 0 \)[/tex]).
We have the result: [tex]\( c \approx 97.81 \)[/tex]
3. Match with Given Options:
Now, compare these results to the options given:
- A) [tex]\( f(x) = -13.5x + 97.8 \)[/tex]
- B) [tex]\( f(x) = -13.5x + 7.3 \)[/tex]
- C) [tex]\( f(x) = 97.8x - 13.5 \)[/tex]
- D) [tex]\( f(x) = 7.3x - 97.8 \)[/tex]
The closest match based on both the slope and the y-intercept is option A, [tex]\( f(x) = -13.5x + 97.8 \)[/tex].
Therefore, the best function to model the data is option A: [tex]\( f(x) = -13.5x + 97.8 \)[/tex].
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