To find a function \( f(x) \) that represents the average height of trees in feet based on the number of years since they were planted, we can first identify the linear relationship between the years and the average height.
Given Data Points:
- (1, 10)
- (3, 24)
- (6, 45)
- (11, 80)
- (15, 105)
Finding the Slope:
The slope \( m \) of a linear function can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (1, 10) and (15, 105):
- \( y_2 = 105 \), \( y_1 = 10 \)
- \( x_2 = 15 \), \( x_1 = 1 \)
Calculating the slope: \[ m = \frac{105 - 10}{15 - 1} = \frac{95}{14} \approx 6.7857 \]
Finding the y-intercept:
Using the point-slope form of a line \( y = mx + b \) and substituting one of the points to find \( b \). Let's use the point (1, 10):
\[ 10 = \left(\frac{95}{14}\right)(1) + b \] \[ b = 10 - \frac{95}{14} = 10 - 6.7857 \approx 3.2143 \]
Function Representation:
Combining slope and y-intercept, we can represent the function as: \[ f(x) = \frac{95}{14}x + \frac{45}{14} \]
Final Simplified Function:
To express \( f(x) \) in a clearer form: \[ f(x) = \frac{95}{14}x + \frac{45}{14} \]
Conclusion:
Thus, the function that represents this situation is: \[ f(x) = \frac{95}{14}x + \frac{45}{14} \] Where \( f(x) \) is the average height of trees in feet, and \( x \) is the number of years since the trees were planted.