Inment A 600-gallon tank is initially full of water and is being drained at a rate of 30 gallons per minute. Complete parts (a) through (d) below. Question $$ 46,2.4 .19 $$ The $$ W $$-intercept is $$ (0,600 $$ of 8 (Type an ordered pair.) Interpret the $$ W $$-intercept. Choose the correct answer below. $$ 80.25 \%, 47.35 $$ of 59 points A. The $$ W $$-intercept is the time it takes to empty the tank. B. The $$ \mathbf{W} $$-intercept is the constant rate of change of water in the tank. C. The $$ W $$-intercept is the initial amount of water in the tank. (d) Find the domain of $$ W $$. Choose the correct domain of $$ W $$ below. A. $$ D=\{t \mid \leq t \leq \square\} $$ B. All real numbers.

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UpStudy AI · Accepted Answer

**Step 1. Analyze the given problem**

A 600-gallon tank is initially full of water and is being drained at a rate of 30 gallons per minute.

The relationship between water remaining $$W$$ (in gallons) and time $$t$$ (in minutes) is modeled by:
$$
W(t) = 600 - 30t
$$
where $$t = 0$$ corresponds to the initial time.

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**Step 2. Interpret the $$W$$-intercept**

The $$W$$-intercept is the point where $$t = 0$$. Plugging $$t = 0$$ into the equation:
$$
W(0) = 600 - 30(0) = 600
$$
Thus, the $$W$$-intercept is $$(0,600)$$.

This represents the initial amount of water in the tank.

The correct interpretation is:  
C. The $$W$$-intercept is the initial amount of water in the tank.

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**Step 3. Find the domain of $$W$$**

The domain of the function $$W(t)$$ reflects the possible values of time $$t$$. Since we are draining the tank starting at $$t=0$$ until the tank is empty, we need to find when the tank becomes empty:
$$
600 - 30t = 0 \quad \Rightarrow \quad 30t = 600 \quad \Rightarrow \quad t = \frac{600}{30} = 20
$$
Thus, the tank is empty at $$t = 20$$ minutes.

Since the model only makes sense from when the process starts until the tank is empty, the domain is:
$$
0 \leq t \leq 20
$$

Thus, the correct domain is:
A. $$D=\{t \mid 0 \leq t \leq 20\}$$

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**Final Answers:**

- The $$W$$-intercept is $$(0,600)$$, which means the initial amount of water in the tank.
- The domain of $$W$$ is $$\{t \mid 0 \leq t \leq 20\}$$.
- The $$W$$-intercept is $$(0,600)$$, indicating the initial amount of water in the tank. - The domain of $$W$$ is $$0 \leq t \leq 20$$.

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