Solution: Given $ v(t) = t^2 - 4t + mt $ and $ v(2) = 8 $, substitute $ t = 2 $:

Title: How to Solve for $ m $ in the Velocity Function $ v(t) = t^2 - 4t + mt $ Using the Condition $ v(2) = 8 $

When working with mathematical models in physics or engineering, functions like $ v(t) $ represent velocity over time. In this article, we’ll explore how to solve for an unknown parameter — $ m $ — in the velocity function $ v(t) = t^2 - 4t + mt $, using a given condition: $ v(2) = 8 $. Substituting $ t = 2 $ is a key step in determining $ m $, and this technique is fundamental in both algebra and applied mathematics.

Understanding the Context

What is the Problem?

We are given the velocity function:
$$
v(t) = t^2 - 4t + mt
$$
and the condition:
$$
v(2) = 8
$$
Our goal is to find the value of $ m $ that satisfies this condition.

Step 1: Substitute $ t = 2 $ into the Function

Solution: Given $ v(t) = t^2 - 4t + mt $ and $ v(2) = 8 $, substitute $ t = 2 $:

Key Insights

To evaluate $ v(2) $, substitute $ t = 2 $ into the expression for $ v(t) $:
$$
v(2) = (2)^2 - 4(2) + m(2)
$$
Simplify each term:
$$
v(2) = 4 - 8 + 2m
$$
$$
v(2) = -4 + 2m
$$

Step 2: Apply the Given Condition

We know $ v(2) = 8 $. So set the expression equal to 8:
$$
-4 + 2m = 8
$$

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Final Thoughts

Step 3: Solve for $ m $

Add 4 to both sides:
$$
2m = 12
$$
Now divide both sides by 2:
$$
m = 6
$$

Why This Matters: Application in Real Problems

This method of substituting a known input to solve for a parameter is widely used across disciplines. For example:

In physics, when modeling motion, constants like $ m $ may represent mass or resistance factors.
In economics or optimization, parameters often encode real-world constraints.
Solving $ v(2) = 8 $ confirms that $ m = 6 $ ensures the model matches observed data at time $ t = 2 $, validating the equation’s accuracy.

Final Answer

By substituting $ t = 2 $ into $ v(t) = t^2 - 4t + mt $ and applying $ v(2) = 8 $, we determined:
$$
m = 6
$$
Thus, the fully defined velocity function is:
$$
v(t) = t^2 - 4t + 6t = t^2 + 2t
$$
and the condition is satisfied.

Summary Checklist

✔ Substitute $ t = 2 $ into $ v(t) = t^2 - 4t + mt $
✔ Simplify to $ -4 + 2m $
✔ Set equal to 8 and solve: $ 2m = 12 $ → $ m = 6 $
✔ Confirm model accuracy with real-world or analytical context