First, we need to determine a linear equation that models the given data. The data consists of points as follows:
- (3, 25)
- (6, 85)
- (9, 168)
- (12, 205)
We can choose two points to find the slope of the line. Let's take the first two points: (3, 25) and (6, 85).
The formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from the points:
\[ m = \frac{85 - 25}{6 - 3} = \frac{60}{3} = 20 \]
Now we will use the point-slope form of the equation of a line, which is given by \( y - y_1 = m(x - x_1) \). We'll use the first point (3, 25):
\[ y - 25 = 20(x - 3) \]
Expanding this:
\[ y - 25 = 20x - 60 \]
Adding 25 to both sides:
\[ y = 20x - 35 \]
Now, we can use this linear model to predict the number of bacteria after 24 hours. Substitute \( x = 24 \) into the equation:
\[ y = 20(24) - 35 \] \[ y = 480 - 35 \] \[ y = 445 \]
However, since this number does not match any of the choices provided (35, 277, 463, 819), the linear model may not be the best fit for the data.
To consider an adjustment or better fit, let's also examine our original data more closely. Given that the growth appears to be more than linear, a quadratic function might actually be more appropriate, but based on the provided answers, it could hint towards an exponential growth model.
If I re-check the values for growth per time interval assuming a linear approximation just for easier prediction, we can see:
- From 3 to 6 hours: Increase of 60 bacteria
- From 6 to 9 hours: Increase of 83 bacteria
- From 9 to 12 hours: Increase of 37 bacteria
The increases themselves do not follow a fixed pattern indicating linearity.
However, based on the most recent data point (at 12 hours having 205), and projecting an average growth rate observed, trying to just interpolate neatly from existing data as everything generally goes upwards indicates that the value after 24 hours could come from evaluation or check.
From experimental results or curve fit might show trends or net results towards the higher counts, especially post 12 hours if extrapolated non-linearly for rapid bacteria growth within certain growth phase metrics of parts observed.
Given all factors considered and possible expected model fits here:
The logical derived guess could rest closest to \( \textbf{463} \).
So, the closest probable prediction for the number of bacteria after 24 hours based on options would be:
463.
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