A tank with some water in it begins to drain. The function v ( t )

The function given is $v(t) = 50 - 3.5t$, where $v(t)$ represents the volume of water in gallons in the tank at time $t$ (in minutes) after the draining process starts.

We will interpret each of the options:

1. The number of gallons of water in the tank when it starts draining: This refers to the initial volume of water at $t = 0$. When $t = 0$, the function becomes $v(0) = 50 - 3.5(0) = 50$. So, the initial amount of water in the tank is 50 gallons. This statement is true.

2. The weight of the tank when it is empty: This option is irrelevant to the function because the function $v(t)$ only describes the volume of water in the tank over time, not the weight of the tank itself. This statement is false.

3. How many minutes it takes for all of the water to drain from the tank: To determine when the tank is empty, we set $v(t) = 0$ and solve for $t$: $50 - 3.5t = 0 \implies 3.5t = 50 \implies t = \frac{50}{3.5} = 14.29 \, \text{minutes}.$ So, it takes about 14.29 minutes for all the water to drain from the tank. However, this is a description of the $t$-coordinate, not the $v$-coordinate. This statement is false for $v$-coordinate.

Summary of correct answers:

• The number of gallons of water in the tank when it starts draining (True)

Would you like further details or have any questions?

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Related Questions:

1. What does the slope $-3.5$ in the equation $v(t) = 50 - 3.5t$ represent?
2. At what time will the tank be half-empty?
3. How does the rate of change of water volume over time affect the draining process?
4. If the initial volume was different, how would that change the equation?
5. What would the function look like if water was draining at a different rate?

Tip: Whenever a linear function describes a physical process, the slope often represents the rate of change per unit of time, and the intercept gives the initial value.