3). Suppose a company is designing a water filtration system. The time in minutes required to fiter
gallons of water is modeled by the function:

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Mar 27,2025

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3). Suppose a company is designing a water filtration system. The time in minutes required to fiter $$ x $$ gallons of water is modeled by the function: $$ T(x)=\frac{200+5 x}{x} $$

UpStudy AI · Accepted Answer

**Step 1. Simplify the function**

The given function is

$$
T(x)=\frac{200+5x}{x}.
$$

We can separate the terms in the numerator:

$$
T(x)=\frac{200}{x} + \frac{5x}{x} = \frac{200}{x} + 5.
$$

**Step 2. Determine the Domain**

Since $$ x $$ represents gallons of water, and we cannot filter $$ 0 $$ gallons, we require $$ x>0 $$. Also, the expression $$\frac{200}{x}$$ is undefined when $$ x=0 $$. Thus, the domain is

$$
x>0.
$$

**Step 3. Analyze the Behavior of the Function**

- **Behavior for Small Values of $$ x $$:**

When $$ x $$ is close to $$ 0 $$ (but positive), $$\frac{200}{x}$$ becomes very large. Hence, $$ T(x) $$ is very large for small $$ x $$.

- **Behavior for Large Values of $$ x $$:**

As $$ x \to \infty $$, the term $$\frac{200}{x}$$ approaches $$ 0 $$ and

$$
  \lim_{x\to\infty} T(x)=0+5=5.
  $$

This indicates that for very large gallons, the time required approaches $$ 5 $$ minutes.

**Step 4. Interpret the Meaning in the Context**

The model

$$
T(x)=\frac{200}{x}+5
$$

shows that there is a fixed time of $$ 5 $$ minutes (perhaps representing setup or processing time) plus a time component $$ \frac{200}{x} $$ that decreases as the number of gallons increases. In practical terms, filtering a larger volume results in a smaller additional time per gallon.

**Step 5. Evaluate the Function for Specific Values (Example)**

- For $$ x=100 $$ gallons:

$$
  T(100)=\frac{200}{100}+5 = 2+5=7 \text{ minutes}.
  $$

- For $$ x=200 $$ gallons:

$$
  T(200)=\frac{200}{200}+5 = 1+5=6 \text{ minutes}.
  $$

- For $$ x=400 $$ gallons:

$$
  T(400)=\frac{200}{400}+5 = 0.5+5=5.5 \text{ minutes}.
  $$

These evaluations further illustrate that as $$ x $$ increases, $$ T(x) $$ gets closer to $$ 5 $$ minutes.

**Conclusion**

The function $$ T(x)=\frac{200+5x}{x} $$ simplifies to

$$
T(x)=\frac{200}{x}+5,
$$

with the domain $$ x>0 $$. It models the filtration time such that for small values of $$ x $$ the time is high, and as $$ x \to \infty $$, the time approaches a lower limit of $$ 5 $$ minutes.
The function simplifies to $$ T(x) = \frac{200}{x} + 5 $$ with $$ x > 0 $$. This means the time to filter $$ x $$ gallons of water increases as $$ x $$ decreases, approaching 5 minutes as $$ x $$ becomes very large.

3). Suppose a company is designing a water filtration system. The time in minutes required to fiter
gallons of water is modeled by the function:

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