The product-exchange function for gasoline $$ x $$ and heating oil $$ y $$, in hundreds of gallons per day, is shown below. $$ y=\frac{170,000-140 x}{170+2 x} $$ Sketch the portion of the function in Quadrant I of the graph, and then find the x-intercept and $$ y $$-intercept in order to estimate the maximum quantities of each product that can be produced. What is the maximum quantity of heating oil that can be produced? What is the maximum quantity of gasoline that can be produced? (Round to the nearest whole number as needed.) (Round to the nearest whole number as needed.)

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To determine the maximum quantities of heating oil and gasoline that can be produced, we'll analyze the given product-exchange function:

$$
y = \frac{170,000 - 140x}{170 + 2x}
$$

Here:
- $$ x $$ represents gasoline in **hundreds of gallons per day**.
- $$ y $$ represents heating oil in **hundreds of gallons per day**.

### 1. **Finding the $$ y $$-Intercept (Maximum Heating Oil)**

The $$ y $$-intercept occurs when $$ x = 0 $$:

$$
y = \frac{170,000 - 140(0)}{170 + 2(0)} = \frac{170,000}{170} = 1,000
$$

**Interpretation:**
- **Maximum Heating Oil**: $$ y = 1,000 $$ hundreds of gallons per day.
- **Total:** $$ 1,000 \times 100 = 100,000 $$ gallons per day.

### 2. **Finding the $$ x $$-Intercept (Maximum Gasoline)**

The $$ x $$-intercept occurs when $$ y = 0 $$:

$$
0 = \frac{170,000 - 140x}{170 + 2x}
$$

Solving for $$ x $$:

$$
170,000 - 140x = 0 \
140x = 170,000 \
x = \frac{170,000}{140} \approx 1,214.29
$$

**Interpretation:**
- **Maximum Gasoline**: $$ x \approx 1,214 $$ hundreds of gallons per day (rounded to the nearest whole number).
- **Total:** $$ 1,214 \times 100 = 121,400 $$ gallons per day.

### 3. **Summary of Maximum Quantities**

- **Maximum Heating Oil Produced:** **100,000 gallons per day**
- **Maximum Gasoline Produced:** **121,400 gallons per day**

### 4. **Graphical Representation**

To sketch the portion of the function in Quadrant I:

- **Plot the $$ y $$-intercept** at $$ (0, 1,000) $$.
- **Plot the $$ x $$-intercept** at $$ (1,214, 0) $$.
- **Draw the curve** that approaches these intercepts, ensuring it remains in Quadrant I (i.e., both $$ x $$ and $$ y $$ are positive).

![Graph of y = (170000 - 140x)/(170 + 2x)](https://i.imgur.com/yourGraphLink.png) *(Note: Replace with an actual graph if needed.)*

### **Final Answers:**

- **Maximum Heating Oil:** 100 000 gallons per day
- **Maximum Gasoline:** 121 400 gallons per day
The maximum quantity of heating oil that can be produced is 100,000 gallons per day, and the maximum quantity of gasoline that can be produced is 121,400 gallons per day.

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