The table of values below represents a linear function and shows the amount of snow that has fallen since a snowstorm began. What is the rate of change?
| Length of Snowfall (hours) | Amount of Snow on the Ground (inches) |
|---|---|
| 0 | 3.30 |
| 0.5 | 4.5 |
| 1.0 | 5.7 |
| 1.5 | 6.9 |
| 2.0 | 8.1 |
A. 1.2 inches per hour
B. 2.4 inches per hour
Answer (1)
Identify two points from the table: (0, 3.30) and (0.5, 4.5).
Calculate the rate of change using the formula: r a t e = x 2 − x 1 y 2 − y 1 .
Substitute the values: r a t e = 0.5 − 0 4.5 − 3.30 = 0.5 1.2 .
The rate of change is 2.4 inches per hour.
Explanation
Understanding the Problem We are given a table of values representing a linear function that shows the amount of snow that has fallen since a snowstorm began. We need to find the rate of change, which is the amount of snow that falls per hour.
Choosing Points and Defining Rate of Change To find the rate of change, we can use any two points from the table. Let's choose the points (0, 3.30) and (0.5, 4.5). The rate of change is the change in the amount of snow divided by the change in time.
Stating the Formula The formula for the rate of change (slope) is: r a t e = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the two points we've chosen.
Calculating the Rate of Change Substituting the points (0, 3.30) and (0.5, 4.5) into the formula, we get: r a t e = 0.5 − 0 4.5 − 3.30 r a t e = 0.5 1.2 r a t e = 2.4
Final Answer The rate of change is 2.4 inches per hour.
Examples
Understanding the rate of change is useful in many real-world situations. For example, if you're tracking the water level in a tank as it fills, the rate of change tells you how quickly the tank is filling up. Similarly, if you're monitoring the temperature of an object as it cools down, the rate of change tells you how quickly the object is losing heat. In this case, knowing the rate of snowfall helps in planning activities and preparing for the weather conditions. If the rate of snowfall is constant, we can predict how much snow will accumulate over a certain period of time, which is crucial for safety and logistical planning.
