Question

At time t=0 water begins to drip out of a pipe into an empty bucket. After 56 minutes 8 inches of water are in the bucket. What linear function in the form y=mx+b represents the amount of water in inches,w, in the bucket after t minutes

Answer 1

The linear function of the bucket is
`y = (1)/(7)\cdot x`, where
`y` represents the amount of water, measured in inches, and
`x` is the time, measured in minutes.

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

`A(x,y) = (0\,min, 0\,in)`

Final condition of the bucket

`B(x,y) = (56\,min, 8 in)`

The equation of the line is defined by the following model:

`y = m\cdot x + b` (1)

Where:

`x` - Independent variable, dimensionless.

`y` - Dependent variable, dimensionless.

`b` - y-Intercept, dimensionless.

`m` - Slope, dimensionless.

Based on the known conditions of the bucket, we obtain the following system of linear equations:

`b = 0` (2)

`56\cdot m +b = 8` (3)

The solution of the system of equations is:

`m = (1)/(7)` and
`b = 0`

Then, the linear function of the bucket is
`y = (1)/(7)\cdot x`, where
`y` represents the amount of water, measured in inches, and
`x` is the time, measured in minutes.