Verified Answer :
- Start with the linear form \(d(t) = mt + b\), where \(m\) is the speed and \(b\) is the initial distance.
- Use the condition \(d(0) = 12\) to find that the y-intercept is \(b = 12\).
- Use the condition \(d(3) = 21\) to solve for the slope: \(3m + 12 = 21\), which results in \(m = 3\).
- State the final linear equation: \(\boxed{d(t) = 3t + 12}\).
Explanation
- Analysis of the Problem Data
We are given that Carlos is jogging at a constant speed, meaning the distance \(d\) he is from his starting position at time \(t\) can be modeled by a linear equation of the form \(d(t) = mt + b\), where \(m\) represents his speed (constant rate of change of distance) and \(b\) represents his initial distance from the starting position at time \(t=0\). We are given two specific data points to help us determine these constants: at \(t=0\), the distance is \(d=12\), and at \(t=3\), the distance is \(d=21\).
- Determining the Initial Distance Constant
We start with the general linear model \(d(t) = mt + b\). Using the first condition, where Carlos is \(12\) feet from the starting position at \(t=0\), we substitute these values into the equation:
\[d(0) = m(0) + b = 12\]
This simplifies directly to \(b = 12\). Thus, the initial distance from the starting position is \(12\) feet.
- Solving for the Speed Constant
Now that we know \(b=12\), our equation is \(d(t) = mt + 12\). We use the second condition, which states that after \(t=3\) seconds, Carlos is \(21\) feet from his starting position. Substituting these values into our equation:
\[d(3) = m(3) + 12 = 21\]
To solve for the speed \(m\), we subtract \(12\) from both sides of the equation:
\[3m = 21 - 12\]
\[3m = 9\]
Dividing by \(3\), we find the speed:
\[m = \frac{9}{3} = 3\]
So, Carlos is running at a constant speed of \(3\) feet per second.
- Finalizing the Linear Equation
With the constants determined, we have \(m=3\) and \(b=12\). Substituting these values back into our general form \(d(t) = mt + b\), we arrive at the linear equation representing his distance from the starting point over time:
\[d(t) = 3t + 12\]
This equation shows that for every second that passes, Carlos increases his distance from the starting position by \(3\) feet, starting from an initial distance of \(12\) feet. The final linear equation is \(\boxed{d(t) = 3t + 12}\).
Examples
Linear equations like this are incredibly useful for predicting future positions or outcomes based on current trends. For example, if you know your current speed while biking and your distance from home, you can use a linear equation to estimate when you will arrive at a specific destination. This simple mathematical model acts as a reliable tool for planning journeys, managing resources, or even predicting costs over time, ensuring you stay on track with your goals.