Problem 59 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. A rock is dropped from the top of a 400 -foot cliff. After 1 second, the rock is traveling 32 feet per second. After 3 seconds, the rock is traveling 96 feet per second. a. Assume that the relationship between time and speed is linear and write an equation describing this relationship. Use ordered pairs of the form (time, speed). b. Use this equation to determine the speed of the rock 4 seconds after it is dropped.

Step-by-Step Solution

Answer

The equation is \( y = 32x \); at 4 seconds, the speed is 128 feet per second.

Step 1: Identify the Ordered Pairs

First, we need to recognize the data points given in the problem. The speed is given 1 second and 3 seconds after the rock is dropped. We can represent these data points as ordered pairs: (1, 32) and (3, 96), where the first element is time in seconds and the second is speed in feet per second.

Step 2: Determine the Slope

The slope (m) of the line represents the rate of change of speed with respect to time. It can be calculated using the formula for the slope \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Here, \( y_2 = 96 \), \( y_1 = 32 \), \( x_2 = 3 \), and \( x_1 = 1 \). So, \[m = \frac{{96 - 32}}{{3 - 1}} = \frac{{64}}{{2}} = 32.\]

Step 3: Write the Equation in Slope-Intercept Form

The equation of a line in slope-intercept form is \( y = mx + b \), where m is the slope and b is the y-intercept. We have already calculated m = 32. Now, we'll use one of the points, say (1, 32), to find b. Substituting into the equation:\[32 = 32(1) + b\]\[32 = 32 + b\]\[b = 0\]Thus, the equation is \( y = 32x + 0 \), or simply \( y = 32x \).

Step 4: Use the Equation to Find the Speed After 4 Seconds

Now, we use the equation \( y = 32x \) to find the speed after 4 seconds (x = 4). Substitute x = 4 into the equation:\[y = 32(4) = 128\]So, the speed of the rock after 4 seconds is 128 feet per second.

Key Concepts

Linear Equations

A linear equation is a fundamental mathematical expression, describing a straight line when graphed on a coordinate plane.
This equation is commonly represented in the slope-intercept form: \( y = mx + b \).
Here, \( m \) is the slope, and \( b \) is the y-intercept.Linear equations are widely used to model relationships where there is a consistent rate of change between two variables.

• If the relationship between the variables is constant, the graph of the equation will depict a straight line.
• The slope \( m \) illustrates the rate at which one variable changes with respect to another.
• The y-intercept \( b \) represents the value of \( y \) when \( x \) is zero.

In our rock example, speed is a linear function of time.
The equation \( y = 32x \) indicates that for every second that passes, the speed increases by 32 feet per second.These equations help students and professionals make predictions by providing a simple representation of trends.

Ordered Pairs

Ordered pairs are pairs of numbers used to denote a particular point on a coordinate plane.
They follow the format \((x, y)\), where \( x \) represents the horizontal position and \( y \) the vertical position.In the context of the exercise:

• The ordered pairs are \((1, 32)\) and \((3, 96)\).
• Here, \( x \) corresponds to 'time' measured in seconds, and \( y \) corresponds to 'speed' in feet per second.

Using these ordered pairs, you can determine the characteristics of the linear relationship between the variables.Ordered pairs are also crucial in forming the foundation for plotting lines and understanding geometric relations.
Each point provides a precise spot on the graph, and connecting points shows the path defined by the linear equation.

Rate of Change

The rate of change describes how one variable changes in relation to another.
In linear relationships, this rate is known as the "slope" of the line, represented by \( m \) in the slope-intercept form \( y = mx + b \).Calculating the slope involves dividing the change in the vertical value (\( y \)) by the change in the horizontal value (\( x \)).
This calculation gives a clear picture of how quickly changes occur between the two variables over a certain interval.For our example:

• The change in speed from 32 feet per second to 96 feet per second is 64.
• The change in time from 1 second to 3 seconds is 2.
• Thus, the rate of change, or slope, is calculated as \( \frac{64}{2} = 32 \).

This information tells us that every additional second increases the speed of the rock by 32 feet per second.Understanding rate of change is vital, as it allows predictions and conclusions about the behavior of dynamic systems.