Problem 59 - Algebra A Combined Function

Question

Solve. See Example 7. A rock is dropped from the top of a 400 -foot building. After 1 second, the rock is traveling 32 feet per second. After 3 seconds, the rock is traveling 96 feet per second. Let \(y\) be the rate of descent and \(x\) be the number of seconds since the rock was dropped. a. Write a linear equation that relates time \(x\) to rate \(y\). [Hint: Use the ordered pairs (1,32) and \((3,96) .]\) b. Use this equation to determine the rate of travel of the rock 4 seconds after it was dropped.

Step-by-Step Solution

Answer

The linear equation is \(y = 32x\). At 4 seconds, the rate is 128 feet/sec.

Step 1: Identify Given Points

We're given two points on the line: \((1, 32)\) and \((3, 96)\). These represent the values of \(x\) (time in seconds) and \(y\) (rate of descent in feet per second).

Step 2: Calculate the Slope

The slope \(m\) of a linear equation can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \((x_1, y_1) = (1, 32)\) and \((x_2, y_2) = (3, 96)\), we get: \[ m = \frac{96 - 32}{3 - 1} = \frac{64}{2} = 32 \] So, the slope is 32.

Step 3: Formulate the Linear Equation

Using the slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. We know \(m = 32\), and using the point \((1, 32)\), substitute into the equation: \[ 32 = 32(1) + b \] Solving for \(b\), we get: \[ b = 0 \] Therefore, the equation is \(y = 32x\).

Step 4: Determine the Rate at 4 Seconds

Substitute \(x = 4\) into the equation \(y = 32x\) to find the rate at 4 seconds: \[ y = 32 \times 4 = 128 \] So, the rate of travel 4 seconds after the rock was dropped is 128 feet per second.

Key Concepts

Slope-Intercept Form

Linear equations are often expressed in what is called the slope-intercept form. This is a straightforward way to describe a line with an equation written as \(y = mx + b\). Here, \(m\) represents the slope of the line, which tells us how steep the line is. The variable \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Think of the slope as the rate of change between two variables. It reflects how much \(y\) changes when \(x\) changes. When you know both the slope and y-intercept, you can easily locate any point on the line.

• \(m\) - Slope, demonstrating the change in \(y\) per change in \(x\).
• \(b\) - y-intercept, where the line meets the y-axis.

For the initial exercise, the equation \(y = 32x\) shows a slope of 32 and a y-intercept of 0. This tells us the rate of descent increases by 32 feet per second for every second the rock falls.

Rate of Change

The rate of change in a linear equation is synonymous with the slope, \(m\). It quantifies the relationship between two variables, showing how one variable changes when the other changes.

To calculate the rate of change, we used the formula: \[ m = \frac{y_2-y_1}{x_2-x_1} \]

• Choose two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the \(y\)-values and the \(x\)-values, then divide \((y_2-y_1)\) by \((x_2-x_1)\).

In the problem of the falling rock, the rate of change was calculated as 32 feet per second, indicating that every second, the speed of the rock increases by 32 feet per second. This consistent rate illustrates linearity, justifying why the equation takes the form that it does.

Ordered Pairs

Ordered pairs \((x, y)\) are a way to represent data points on a coordinate plane. Each pair consists of two elements: the first is the value of \(x\), and the second represents the corresponding \(y\) value.

When examining linear relationships, these pairs are crucial as they provide specific points that help in drawing the line on a graph. For example, in the exercise, we were provided with two ordered pairs: \((1, 32)\) and \((3, 96)\). This means:

• At \(x = 1\) second, \(y = 32\) feet per second is the descent rate.
• At \(x = 3\) seconds, \(y = 96\) feet per second.

These pairs were used to calculate the rate of change and helped in determining the slope of the line. Understanding how to use ordered pairs to formulate a linear equation is a fundamental skill when working with linear equations.