Solution: To find when the heights are equal, set $ h_A(t) = h_B(t) $:

["# How to Solve When Heights Are Equal: A Clear Approach Using $ h_A(t) = h_B(t) $", "In mathematical modeling and real-world problem-solving, one common challenge is determining when two quantities remain equal over time. Whether analyzing projectile motion, comparing growth rates, or studying synchronized processes, identifying the exact moment when two heights are equal is crucial. The powerful solution lies in setting the height functions equal: $ h_A(t) = h_B(t) $.", "This method provides a straightforward algebraic approach to finding the time $ t $ when two heights match—offering clarity and precision for students, engineers, scientists, and anyone working with dynamic systems. In this article, we’ll explore how to apply this solution methodically, with examples and practical tips to help you master the concept.", "---", "## Why Set $ h_A(t) = h_B(t) $?", "When two height functions are defined as $ h_A(t) $ and $ h_B(t) $, solving $ h_A(t) = h_B(t) $ turns the equality task into an equation that can be manipulated algebraically. This step turns a qualitative question—“When do the heights match?”—into a concrete computation involving time $ t $.", "By equating the expressions, we eliminate variability and isolate $ t $. This algebraic equality lets us apply standard techniques—factoring, simplifying, solving linear or quadratic equations—to find exact solutions.", "---", "## Step-by-Step Solution Process", "### Step 1: Define the height functions\nStart by clearly expressing both height functions in terms of time $ t $. For example:\n- $ h_A(t) = -16t^2 + v_0A t + h_0A $ (projectile motion with initial velocity and height)\n- $ h_B(t) = -16t^2 + v_0B t + h_0B $ (another trajectory or object with different parameters)", "### Step 2: Set the equations equal\nWrite:\n$$\n-16t^2 + v_0A t + h_0A = -16t^2 + v_0B t + h_0B\n$$", "### Step 3: Simplify the equation\nSubtract $ -16t^2 $ from both sides to cancel quadratic terms:\n$$\nv_0A t + h_0A = v_0B t + h_0B\n$$", "### Step 4: Collect like terms\nBring all terms involving $ t $ to one side and constants to the other:\n$$\n(v_0A - v_0B)t = h_0B - h_0A\n$$", "### Step 5: Solve for $ t $\nIf $ v_0A <br/>\ne v_0B $, then:\n$$\nt = \frac{h_0B - h_0A}{v_0A - v_0B}\n$$\nThis gives the exact time(s) when the heights are equal.", "---", "## Example: Projectile Heights Equals", "Let:\n- $ h_A(t) = -16t^2 + 32t + 5 $\n- $ h_B(t) = -16t^2 + 20t + 9 $", "Set equal:\n$$\n-16t^2 + 32t + 5 = -16t^2 + 20t + 9\n$$\nCancel $ -16t^2 $:\n$$\n32t + 5 = 20t + 9\n$$\nSolve:\n$$\n12t = 4 \quad \Rightarrow \quad t = \frac{1}{3}\n$$", "So, the heights are equal at $ t = \frac{1}{3} $ seconds.", "---", "## Practical Tips for Real-World Use", "- Check for valid solutions: Ensure $ t \geq 0 $ since negative time may not be physically meaningful in contexts like motion.\n- Multiple solutions: If the equation simplifies to a quadratic or higher, there may be zero, one, or two valid times when heights match—analyze discriminants and check domain.\n- Graphical verification: Plotting the functions confirms solution accuracy and reveals behavioral insights.\n- Units matter: Ensure time variable $ t $ uses consistent units (seconds) for correct interpretation.", "---", "## Summary", "Setting $ h_A(t) = h_B(t) $ is a fundamental and efficient solution method to find when two heights are equal. By equating the expressions, simplifying algebraically, and solving for $ t $, this technique unlocks clear timing insights across engineering, physics, economics, and more. Whether you’re modeling projectile paths, comparing investment growth, or studying biological development, this approach forms a powerful analytical foundation.", "Key takeaway: When heights (or any two dynamic quantities) are equal, solve $ h_A(t) = h_B(t) $—it’s fast, reliable, and mathematically elegant."]