Final answer:
To model the growth of the vine maple, we calculate the slope of the linear equation with the given points (4, 9.33) and (7, 13.33), finding a slope of 4/3. The y-intercept is determined to be 4 by substituting one of the points into our slope-intercept equation. Thus, the model of the vine maple's growth as a function of time is y = (4/3)x + 4.
Step-by-step explanation:
We are given that a vine maple grows linearly, and we have two data points for its growth: at four weeks it is 9.33 inches tall, and at seven weeks it is 13.33 inches tall. To write a linear equation that models the growth of the vine maple as a function of time, we can use the two given data points to find the slope (m) and y-intercept (b) of the line.
First, we find the slope (rate of growth per week) using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are our data points:
m = (13.33 - 9.33) / (7 - 4) = 4 / 3
The vine maple grows at a rate of 4/3 inches per week. Now we need to find the y-intercept, which we can do by substituting one of the points into the equation y = mx + b. Let's use the first point (4, 9.33):
9.33 = (4/3) * 4 + b
9.33 = 16/3 + b
b = 9.33 - 16/3
b = 28/3 - 16/3
b = 12/3
b = 4
The y-intercept (b) is 4, which represents the height of the vine maple at the time it was planted.
Therefore, the equation that models the growth of the vine maple in inches (y) as a function of time in weeks (x) is:
y = (4/3)x + 4