Problem 54 - Algebra 1

Question

Find the rate of change between the two points. Give the units of measure for the rate. \((53,44)\) and \((32,14) ; x\) in seconds, \(y\) in liters.

Step-by-Step Solution

Answer

The rate of change or slope between the two points is \(10/7\) litres per second.

Step 1: Identify the coordinates

Identify the coordinates of the two points. The first point is (53,44) and the second point is (32,14). It means the first point at time 53 seconds, the volume is 44 liters and at time 32 seconds, the volume is 14 liters.

Step 2: Calculate the differences

Calculate the differences in both the x-coordinates and the y-coordinates. The difference in x-coordinates \(Δx\) is \(53 - 32 = 21\) seconds. The difference in y-coordinates \(Δy\) is \(44 - 14 = 30\) liters.

Step 3: Compute the rate of change

Compute the rate of change which is the slope of the line passing through the two points, by dividing the difference in y-coordinates by the difference in x-coordinates. \(Rate = Δy / Δx = 30 litres / 21 seconds = 10 / 7 litres/second.\)

Key Concepts

Coordinate Points

When discussing the rate of change, coordinate points serve as a foundation. A coordinate point in a two-dimensional plane is represented as \(x, y\). It consists of two components:

• The x-coordinate, which typically stands for a specific point in time or another independent variable.
• The y-coordinate, representing a dependent variable, like a measurement or count, at that specific point in time.

For example, in our exercise with the points \(53, 44\) and \(32, 14\):

• The first set of coordinates, (53, 44), tells us that at 53 seconds, there was a measurement of 44 liters.
• Similarly, the second set, (32, 14), indicates that at 32 seconds, the measurement was 14 liters.

Understanding these points helps define the changes and trends between variables, which is crucial for calculating the rate of change.

Slope Calculation

The slope calculation is a key step in understanding how quickly or slowly one variable changes in relation to another. In our problem, the slope is akin to the rate of change. It's computed by considering the differences in both the x and y coordinates:

• First, calculate \( \Delta x\), the change in x-coordinates. For our points, \( \Delta x = 53 - 32 = 21\) seconds.
• Next, find \( \Delta y\), the change in y-coordinates. Here, \( \Delta y = 44 - 14 = 30\) liters.

The slope (or rate of change) is the division of these two values: \[ \text{Slope} = \frac{\Delta y}{\Delta x} = \frac{30 \text{ liters}}{21 \text{ seconds}} = \frac{10}{7} \text{ liters/second}\].
This result means that for every additional second, the quantity increases by \frac{10}{7}\ liters.
Understanding slope helps in predicting outcomes and analyzing data trends.

Units of Measure

Units of measure are crucial in interpreting the rate of change correctly, as they provide the context and scale of the data. In our problem, the units help clarify what the rate represents:

• The x-values, or time intervals, are measured in seconds. This gives us the temporal scale of our data.
• The y-values, symbolizing a variable quantity, are measured in liters. This indicates that we're dealing with volume.
• Thus, when we find the rate of change as \[ \frac{10}{7} \text{ liters/second} \], it tells us how much the volume (in liters) changes per second.

By paying attention to the units of measure, we are equipped to accurately interpret the meaning and application of our calculation results, ensuring that our answers are both meaningful and precise in real-world contexts.