Rate of Change and Slope Answer | PDF | Slope | Mathematical Objects - Free Printable

You're looking at a worksheet from a math textbook (Glencoe Algebra 2, Chapter 2) that focuses on Average Rate of Change.

The main concept is:
> Average Rate of Change = (Change in Output) / (Change in Input)
> or more formally:
> (f(b) - f(a)) / (b - a)

This is essentially the slope of the line connecting two points on a graph or in a table. It tells you how much the dependent variable (output) changes, on average, for each unit change in the independent variable (input).

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Let’s solve each exercise step-by-step:

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Exercise 1

Data:
| Time (min) | Height (ft) |
|------------|-------------|
| 0 | 0 |
| 1 | 41 |
| 2 | 82 |
| 3 | 123 |
| 4 | 164 |

We need to find the rate of change for each set of data — meaning between each consecutive pair of points.

- From t=0 to t=1:
`(41 - 0) / (1 - 0) = 41 / 1 = 41 ft/min`

- From t=1 to t=2:
`(82 - 41) / (2 - 1) = 41 / 1 = 41 ft/min`

- From t=2 to t=3:
`(123 - 82) / (3 - 2) = 41 / 1 = 41 ft/min`

- From t=3 to t=4:
`(164 - 123) / (4 - 3) = 41 / 1 = 41 ft/min`

✔ Answer: 41 feet per minute (constant rate)

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Exercise 2

Data:
| Time (min) | Altitude (ft) |
|------------|---------------|
| 0 | 0 |
| 2 | 1,200 |
| 4 | 2,400 |
| 6 | 3,600 |
| 8 | 4,800 |

Again, calculate rate of change between consecutive points.

- From t=0 to t=2:
`(1200 - 0) / (2 - 0) = 1200 / 2 = 600 ft/min`

- From t=2 to t=4:
`(2400 - 1200) / (4 - 2) = 1200 / 2 = 600 ft/min`

- From t=4 to t=6:
`(3600 - 2400) / (6 - 4) = 1200 / 2 = 600 ft/min`

- From t=6 to t=8:
`(4800 - 3600) / (8 - 6) = 1200 / 2 = 600 ft/min`

✔ Answer: 600 feet per minute (constant rate)

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Exercise 3

Data:
| Vehicles through Tunnel (per min) | Speed (mph) |
|-----------------------------------|-------------|
| 75 | 40 |
| 150 | 35 |
| 225 | 30 |
| 300 | 25 |
| 375 | 20 |

Here, input = vehicles per minute, output = speed (mph). We’re finding how speed changes as traffic increases.

- From 75 to 150 vehicles:
`(35 - 40) / (150 - 75) = (-5) / 75 = -1/15 mph per vehicle per minute`

- From 150 to 225:
`(30 - 35) / (225 - 150) = (-5) / 75 = -1/15`

- From 225 to 300:
`(25 - 30) / (300 - 225) = (-5) / 75 = -1/15`

- From 300 to 375:
`(20 - 25) / (375 - 300) = (-5) / 75 = -1/15`

✔ Answer: -1/15 mph per vehicle per minute
*(Negative means speed decreases as traffic increases)*

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Exercise 4

Data:
| Depth of seismic waves (miles) | Distance from epicenter (miles) |
|-------------------------------|----------------------------------|
| 0 | 0 |
| 0.5 | 2.97 |
| 1 | 5.49 |
| 1.5 | 7.62 |
| 2 | 9.59 |

Input = depth (miles), Output = distance (miles). Find how distance changes with depth.

- From d=0 to d=0.5:
`(2.97 - 0) / (0.5 - 0) = 2.97 / 0.5 = 5.94 miles/mile`

- From d=0.5 to d=1:
`(5.49 - 2.97) / (1 - 0.5) = 2.52 / 0.5 = 5.04`

- From d=1 to d=1.5:
`(7.62 - 5.49) / (1.5 - 1) = 2.13 / 0.5 = 4.26`

- From d=1.5 to d=2:
`(9.59 - 7.62) / (2 - 1.5) = 1.97 / 0.5 = 3.94`

✔ Answer: Rates are not constant — they decrease as depth increases:
→ 5.94, 5.04, 4.26, 3.94 miles per mile

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Exercise 5

Data:
| Water through Channel (liters/sec) | Time (sec) |
|------------------------------------|------------|
| 200 | 0 |
| 210 | 1 |
| 220 | 2 |
| 230 | 3 |
| 240 | 4 |

Input = time, Output = water flow (liters/sec). Find how water flow changes over time.

- From t=0 to t=1:
`(210 - 200) / (1 - 0) = 10 / 1 = 10 L/s per sec`

- From t=1 to t=2:
`(220 - 210) / (2 - 1) = 10 / 1 = 10`

- From t=2 to t=3:
`(230 - 220) / (3 - 2) = 10 / 1 = 10`

- From t=3 to t=4:
`(240 - 230) / (4 - 3) = 10 / 1 = 10`

✔ Answer: 10 liters per second per second
*(This is acceleration of water flow!)*

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Exercise 6

Data:
| Distance between Two Trees (m) | Time (sec) |
|-------------------------------|------------|
| 0 | 0 |
| 3.7 | 1 |
| 7.2 | 2 |
| 10.5 | 3 |
| 13.6 | 4 |

Input = time, Output = distance. Find how distance changes over time (i.e., average speed).

- From t=0 to t=1:
`(3.7 - 0) / (1 - 0) = 3.7 m/s`

- From t=1 to t=2:
`(7.2 - 3.7) / (2 - 1) = 3.5 / 1 = 3.5 m/s`

- From t=2 to t=3:
`(10.5 - 7.2) / (3 - 2) = 3.3 / 1 = 3.3 m/s`

- From t=3 to t=4:
`(13.6 - 10.5) / (4 - 3) = 3.1 / 1 = 3.1 m/s`

✔ Answer: Rates are decreasing slightly:
→ 3.7, 3.5, 3.3, 3.1 meters per second

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## ✔ Final Answers Summary:

1. 41 feet per minute
2. 600 feet per minute
3. -1/15 mph per vehicle per minute
4. Not constant: 5.94, 5.04, 4.26, 3.94 miles per mile
5. 10 liters per second per second
6. Decreasing: 3.7, 3.5, 3.3, 3.1 meters per second

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## 💡 Key Takeaway:

The average rate of change is just slope between two points. If it’s constant across all intervals, the relationship is linear. If it changes, the relationship is nonlinear.

Let me know if you’d like graphs or further explanations!