Answer:
The linear function of the bucket is [tex]y = \frac{1}{7}\cdot x[/tex], where [tex]y[/tex] represents the amount of water, measured in inches, and [tex]x[/tex] is the time, measured in minutes.
Step-by-step explanation:
According to the Euclidean and Analytical Geometries, we can construct a line by knowing two distinct points on a plane. Besides, we know two different conditions for the bucket:
Initial condition of the bucket
[tex]A(x,y) = (0\,min, 0\,in)[/tex]
Final condition of the bucket
[tex]B(x,y) = (56\,min, 8 in)[/tex]
The equation of the line is defined by the following model:
[tex]y = m\cdot x + b[/tex] (1)
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
[tex]b[/tex] - y-Intercept, dimensionless.
[tex]m[/tex] - Slope, dimensionless.
Based on the known conditions of the bucket, we obtain the following system of linear equations:
[tex]b = 0[/tex] (2)
[tex]56\cdot m +b = 8[/tex] (3)
The solution of the system of equations is:
[tex]m = \frac{1}{7}[/tex] and [tex]b = 0[/tex]
Then, the linear function of the bucket is [tex]y = \frac{1}{7}\cdot x[/tex], where [tex]y[/tex] represents the amount of water, measured in inches, and [tex]x[/tex] is the time, measured in minutes.
