Linear Equation: Time Vs. Height | Step-by-Step Solution

Let's dive into how to find the equation that represents the relationship between time and height, given the data you've provided. This is a classic linear equation problem, and we'll break it down step-by-step so it's super easy to follow. Guys, understanding linear equations is crucial in math and many real-world applications, so let's get started!

Understanding Linear Relationships

First off, what exactly is a linear relationship? In simple terms, it's a relationship between two variables that can be represented by a straight line. The general form of a linear equation is y = mx + b, where:

• y is the dependent variable (in our case, height h)
• x is the independent variable (in our case, time t)
• m is the slope of the line (the rate of change of y with respect to x)
• b is the y-intercept (the value of y when x is 0)

In our specific scenario, we want to find an equation in the form of h = mt + b. We're given a table of values that show how the height h changes over time t. To find the equation, we need to determine the slope (m) and the y-intercept (b).

So, how do we figure out these crucial components? The slope tells us how much the height changes for each second that passes, and the y-intercept gives us the starting height at time zero. We'll use the data points from the table to calculate these values. Understanding this foundation is key to cracking the problem, so let's move on to calculating the slope!

Calculating the Slope (m)

The slope, often referred to as m, represents the rate of change in a linear relationship. It tells us how much the dependent variable (height, in our case) changes for every unit change in the independent variable (time). The formula to calculate the slope (m) given two points (t1, h1) and (t2, h2) is:

m = (h2 - h1) / (t2 - t1)

Let's pick two points from our table to make this calculation. We can use the first two points: (0, 4000) and (5, 3500). Plugging these values into the formula, we get:

m = (3500 - 4000) / (5 - 0) = -500 / 5 = -100

This means that for every second that passes, the height decreases by 100 feet. The negative sign indicates a decreasing relationship, which makes sense in our scenario as the object is descending. Isn't it cool how a single number can tell us so much about what's happening? Now that we've nailed down the slope, let's move on to finding the y-intercept, which will give us the starting point of our descent.

Determining the y-intercept (b)

The y-intercept, denoted as b, is the point where the line crosses the y-axis. In the context of our problem, it represents the height at time t = 0. Looking at the data table, we can see that when t = 0, h = 4000 feet. So, the y-intercept (b) is simply 4000.

Sometimes, you might not have the y-intercept explicitly given in the data. In that case, you can use the slope-intercept form of the linear equation (h = mt + b) and plug in the slope (m) and the coordinates of any point (t, h) from the table to solve for b. For instance, if we didn't know the y-intercept, we could use the point (5, 3500) and our calculated slope of -100:

3500 = -100 * 5 + b 3500 = -500 + b b = 3500 + 500 = 4000

As you can see, we arrive at the same y-intercept value. Knowing the y-intercept is crucial because it anchors our linear equation. It tells us the initial condition before time starts ticking. With both the slope and y-intercept in hand, we're ready to assemble the complete linear equation!

Constructing the Linear Equation

Now that we have both the slope (m) and the y-intercept (b), we can finally put together the linear equation that represents the relationship between time and height. Remember the slope-intercept form of a linear equation: h = mt + b.

We calculated the slope m to be -100, and we determined the y-intercept b to be 4000. Plugging these values into the equation, we get:

h = -100t + 4000

This equation tells us the height (h) at any given time (t). For example, if we want to find the height at t = 12 seconds, we simply plug in 12 for t:

h = -100 * 12 + 4000 = -1200 + 4000 = 2800

So, at 12 seconds, the height would be 2800 feet. How cool is that? We've built a mathematical model that allows us to predict the height at any point in time. This equation not only represents the data we were given but also provides a powerful tool for understanding the dynamics of the situation. Now, let's take a moment to interpret what this equation means in the real world.

Interpreting the Equation

Our linear equation, h = -100t + 4000, is more than just a bunch of symbols and numbers; it tells a story. Let's break down what each part of the equation means in the context of our problem.

• The height h represents the altitude of the object in feet at a given time.
• The time t represents the elapsed time in seconds.
• The slope m = -100 represents the rate of descent. The negative sign indicates that the object is losing altitude, and the 100 means it's descending at a rate of 100 feet per second. This is a crucial piece of information because it tells us how quickly the object is falling.
• The y-intercept b = 4000 represents the initial height of the object at time t = 0. This is the starting point of our observation, and it's a key reference point for understanding the entire scenario.

So, the equation is essentially saying: