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A tank is being filled with a liquid. The function $V$ gives the volume of liquid in the tank, in liters, after $t$ minutes. $\newline$ What is the best interpretation for the following statement? $\newline$ The value of the derivative of $V$ at $t=1$ is equal to $2$ . $\newline$ Choose $1$ answer: $\newline$ (A) After $1$ minute, the tank was being filled at a rate of $2$ liters. $\newline$ (B) After $1$ minute, the tank had $2$ liters of liquid. $\newline$ (C) After $1$ minute, the tank was being filled at a rate of $2$ liters per minute. $\newline$ (D) During the first minute, the tank was being filled at a rate of $2$ liters per minute.

Full solution

Q. A tank is being filled with a liquid. The function

V

gives the volume of liquid in the tank, in liters, after

t

minutes.

\newline

What is the best interpretation for the following statement?

\newline

The value of the derivative of

V

t=1

is equal to

2

\newline

Choose

1

answer:

\newline

(A) After

1

minute, the tank was being filled at a rate of

2

liters.

\newline

(B) After

1

minute, the tank had

2

liters of liquid.

\newline

1

minute, the tank was being filled at a rate of

2

liters per minute.

\newline

(D) During the first minute, the tank was being filled at a rate of

2

liters per minute.

Interpretation of Derivative: The derivative of $V$ with respect to $t$ represents the rate of change of the volume with respect to time. So, the derivative of $V$ at $t=1$ being equal to $2$ means that at $t=1$ , the rate at which the volume is changing is $2$ liters per unit of time.
Rate Calculation: Since the unit of time given in the problem is minutes, the rate of $2$ liters per unit of time at $t=1$ translates to $2$ liters per minute.
Matching with Choices: Now, we need to match this interpretation with the given choices. Choice (A) says ＂After $1$ minute, the tank was being filled at a rate of $2$ liters,＂ which is incomplete because it doesn't specify the time frame for the rate. Choice (B) says ＂After $1$ minute, the tank had $2$ liters of liquid,＂ which is incorrect because it describes the volume, not the rate. Choice (C) says ＂After $1$ minute, the tank was being filled at a rate of $2$ liters per minute,＂ which correctly describes the rate of change of volume at $t=1$ . Choice (D) says ＂During the first minute, the tank was being filled at a rate of $2$ liters per minute,＂ which is incorrect because it describes the average rate over the first minute, not the specific rate at $t=1$ .