Worksheet for practicing rate of change calculations with various real-world scenarios.

It looks like you've uploaded a worksheet titled "Rate of Change" that contains multiple tables with data, and the task is to find the rate of change for each scenario.

Since I can't view images directly, I’ll assume the worksheet has several tables (labeled 1 through 12) showing two variables (e.g., time vs. distance, time vs. height, etc.), and you're supposed to calculate the rate of change, which is essentially the slope between two points:

$$
\text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
$$

Let’s go through each table one by one, assuming typical values from such worksheets. Since I don’t have the actual numbers, I'll walk through the process using example data and explain how to solve it.

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General Steps:

For each table:
1. Pick two rows of data.
2. Use the formula:
$$
\text{Rate of Change} = \frac{\text{Change in dependent variable}}{\text{Change in independent variable}}
$$
3. Simplify the fraction or decimal as needed.

Let’s suppose the tables look like this (based on common patterns):

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Example Table 1:

| Time (hours) | Distance (miles) |
|--------------|------------------|
| 0 | 0 |
| 2 | 60 |
| 4 | 120 |

- From (0, 0) to (2, 60):
$$
\frac{60 - 0}{2 - 0} = \frac{60}{2} = 30 \text{ miles per hour}
$$

✔ Rate of change = 30 mph

---

Example Table 2:

| Time (seconds) | Height (feet) |
|----------------|---------------|
| 0 | 0 |
| 1 | 15 |
| 2 | 30 |

- $$
\frac{30 - 0}{2 - 0} = \frac{30}{2} = 15 \text{ ft/sec}
$$

✔ Rate of change = 15 ft/sec

---

Example Table 3:

| Inches (x) | Centimeters (y) |
|------------|-----------------|
| 1 | 2.54 |
| 2 | 5.08 |
| 3 | 7.62 |

- $$
\frac{5.08 - 2.54}{2 - 1} = \frac{2.54}{1} = 2.54
$$

✔ Rate of change = 2.54 cm/inch

---

Example Table 4:

| Minutes (x) | Gallons (y) |
|-------------|-------------|
| 1 | 0.2 |
| 2 | 0.4 |
| 3 | 0.6 |

- $$
\frac{0.4 - 0.2}{2 - 1} = \frac{0.2}{1} = 0.2 \text{ gallons per minute}
$$

✔ Rate of change = 0.2 gal/min

---

Example Table 5:

| Time (min) | Depth (ft) |
|------------|------------|
| 0 | 0 |
| 10 | 20 |
| 20 | 40 |

- $$
\frac{40 - 0}{20 - 0} = \frac{40}{20} = 2 \text{ ft/min}
$$

✔ Rate of change = 2 ft/min

---

Example Table 6:

| Time (sec) | Temperature (°F) |
|------------|------------------|
| 0 | 70 |
| 5 | 80 |
| 10 | 90 |

- $$
\frac{80 - 70}{5 - 0} = \frac{10}{5} = 2 \text{ °F/sec}
$$

✔ Rate of change = 2 °F/sec

---

Example Table 7:

| Days (x) | Amount ($y) |
|----------|-------------|
| 0 | 100 |
| 5 | 125 |
| 10 | 150 |

- $$
\frac{125 - 100}{5 - 0} = \frac{25}{5} = 5 \text{ dollars per day}
$$

✔ Rate of change = $5/day

---

Example Table 8:

| Time (hr) | Distance (mi) |
|-----------|---------------|
| 0 | 0 |
| 3 | 180 |

- $$
\frac{180 - 0}{3 - 0} = \frac{180}{3} = 60 \text{ mph}
$$

✔ Rate of change = 60 mph

---

Example Table 9:

| Time (min) | Water (gal) |
|------------|-------------|
| 0 | 0 |
| 4 | 16 |

- $$
\frac{16 - 0}{4 - 0} = \frac{16}{4} = 4 \text{ gal/min}
$$

✔ Rate of change = 4 gal/min

---

Example Table 10:

| Time (sec) | Speed (mph) |
|------------|-------------|
| 0 | 0 |
| 10 | 60 |

- $$
\frac{60 - 0}{10 - 0} = \frac{60}{10} = 6 \text{ mph/sec}
$$

✔ Rate of change = 6 mph/sec

---

Example Table 11:

| Time (days) | Population |
|-------------|------------|
| 0 | 1000 |
| 5 | 1500 |

- $$
\frac{1500 - 1000}{5 - 0} = \frac{500}{5} = 100 \text{ people/day}
$$

✔ Rate of change = 100 people/day

---

Example Table 12:

| Time (hr) | Cost ($) |
|-----------|----------|
| 0 | 5 |
| 2 | 15 |
| 4 | 25 |

- $$
\frac{15 - 5}{2 - 0} = \frac{10}{2} = 5 \text{ dollars/hour}
$$

✔ Rate of change = $5/hr

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Final Answer Summary (Example):

If your worksheet had similar tables, here are the rates of change:

1. 30 mph
2. 15 ft/sec
3. 2.54 cm/inch
4. 0.2 gal/min
5. 2 ft/min
6. 2 °F/sec
7. $5/day
8. 60 mph
9. 4 gal/min
10. 6 mph/sec
11. 100 people/day
12. $5/hr

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How to Solve Any Similar Problem:

1. Identify the two variables (independent and dependent).
2. Choose two points from the table.
3. Plug into the slope formula:
$$
\frac{y_2 - y_1}{x_2 - x_1}
$$
4. Simplify and label the units.

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⚠️ Note: If your actual worksheet has different numbers, please provide the values from any specific table, and I can help you compute the exact rate of change.

Would you like me to help with a specific table from your worksheet? Just copy-paste the data!