Question

the table below represents the amount of water, in gallons, draining from a tank over time.
write the equation of the linear function shown on the graph in slope intercept form.
$y = mx + b$
$y=-50x + 800$
explain what the slope means in the context of this problem.

Explanation:

Step1: Recall the slope - intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Find the \(y\) - intercept

When \(x = 0\) (time \(t=0\) minutes), from the table \(y = 800\) (gallons). So \(b = 800\).

Step3: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,800)\) and \((x_2,y_2)=(9,350)\)

$$m=\frac{350 - 800}{9-0}=\frac{- 450}{9}=-50$$

The slope \(m=-50\) in the equation \(y=-50x + 800\) represents the rate of change of the amount of water in the tank with respect to time. Since the slope is negative, it means that the amount of water in the tank is decreasing. The value of the slope \(-50\) indicates that the amount of water in the tank is decreasing at a rate of \(50\) gallons per minute.

Answer:

The slope \(-50\) means that the amount of water in the tank is decreasing at a rate of \(50\) gallons per minute.