Question

A swimming pool is being drained at a constant rate of 3 inches (depth of the water) per hour. The depth of the water after 5 hours is 32 inches. Write the equation for this function in point slope form

Answer 1

The equation in point slope form is
`y - 47\,in = \left(-3\,(in)/(h)\right)\cdot (t-0\,h)`

Explanation:

Since the swimming pool is being drained at a constant rate, the equation of the process must be a first-order polynomial (linear function), where depth of water decrease as time goes by. The form of the expression is:

`y = m \cdot t + b`

Where:

`t` - Time, measured in hours.

`b` - Initial depth of the water in swimming pool (slope), measured in inches.

`m` - Draining rate, measured in inches per hour.

`y` - Current depth of the water in swimming pool (x-Intercept), measured in inches.

If
`m = -3\,(in)/(h)` and
`y (5\,h) = 32\,in`, the initial depth of the water in swimming pool is:

`b = y - m\cdot t`

`b = 32\,in -\left(-3\,(in)/(h) \right)\cdot (5\,h)`

`b = 47\,in`

The equation in point slope form is:

`y-y_(o) = m \cdot (t-t_(o))`

Where
`y_(o)` and
`t_(o)` are initial depth of the water in swimming pool and initial time, respectively. Then, the equation in point slope form is:

`y - 47\,in = \left(-3\,(in)/(h)\right)\cdot (t-0\,h)`