Question

18 multiple choice 1 point the table below shows the amount of water, y, in gallons, remaining in a tank after it has been draining for x minutes. draining tank minutes (x) gallons in tank (y) 0 500 1 480 2 461 3 439 4 420 the data set was modeled with the linear best - fit function. what is the meaning of the slope of the function? the number of minutes it takes for one gallon of water to drain. the amount of water in the tank after 4 minutes. the initial amount of water in the tank. the average rate, in gallons per minute, that water is draining from the tank.

Explanation:

Step1: Recall the concept of slope in a linear function

In a linear function \(y = mx + b\) (where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope and \(b\) is the \(y\) - intercept), the slope \(m=\frac{\Delta y}{\Delta x}\). Here, \(y\) represents the amount of water in the tank (in gallons) and \(x\) represents the time (in minutes).

Step2: Analyze the units of \(\frac{\Delta y}{\Delta x}\)

The change in \(y\) (\(\Delta y\)) has units of gallons and the change in \(x\) (\(\Delta x\)) has units of minutes. So, \(\frac{\Delta y}{\Delta x}\) has units of \(\frac{\text{gallons}}{\text{minute}}\).

Step3: Eliminate wrong options

• Option 1: The number of minutes it takes for one gallon of water to drain would have units \(\frac{\text{minutes}}{\text{gallon}}\), which is the reciprocal of the slope's units.
• Option 2: The amount of water in the tank after 4 minutes is a value of \(y\) when \(x = 4\), not related to the slope.
• Option 3: The initial amount of water in the tank is the \(y\) - value when \(x = 0\) (i.e., \(b\) in \(y=mx + b\)), not the slope.
• Option 4: Since slope \(m=\frac{\Delta y}{\Delta x}\) and with \(y\) (gallons) and \(x\) (minutes), the slope represents the average rate (in gallons per minute) that water is draining from the tank.

Answer:

The average rate, in gallons per minute, that water is draining from the tank.