Solution: We are given $ h(t) = t^2 - 4t + 5m $ and $ h(3) = 10 $. Substitute $ t = 3 $ into the expression:
["Solution: Unlocking the Value of $ m $ in $ h(t) = t^2 - 4t + 5m $ Given $ h(3) = 10 $", "When working with function-based equations in math and applied sciences, one critical step is verifying known input-output relationships. In this article, we explore how to determine the constant $ m $ in the quadratic function $ h(t) = t^2 - 4t + 5m $, using the condition $ h(3) = 10 $. Substituting $ t = 3 $ into the function is essential to solve for $ m $ and ensure mathematical consistency.", "---", "### Understanding the Function", "The given function is:\n$$\nh(t) = t^2 - 4t + 5m\n$$\nThis represents a quadratic equation in the variable $ t $, with a parameter $ m $ that governs the vertical position of the parabola. Our goal is to determine $ m $ using the specific input-output value $ h(3) = 10 $.", "---", "### Step 1: Substitute $ t = 3 $ into $ h(t) $", "We substitute $ t = 3 $ into the expression for $ h(t) $:\n$$\nh(3) = (3)^2 - 4(3) + 5m\n$$", "Now calculate each term:\n- $ 3^2 = 9 $\n- $ -4 \cdot 3 = -12 $\n- The constant term is $ 5m $", "So the equation becomes:\n$$\nh(3) = 9 - 12 + 5m = -3 + 5m\n$$", "---", "### Step 2: Use the Given Condition", "We are told that $ h(3) = 10 $. Substituting this value gives:\n$$\n-3 + 5m = 10\n$$", "---", "### Step 3: Solve for $ m $", "Add 3 to both sides:\n$$\n5m = 13\n$$", "Divide both sides by 5:\n$$\nm = \frac{13}{5}\n$$", "---", "### Conclusion: The Value of $ m $", "By substituting $ t = 3 $ into $ h(t) = t^2 - 4t + 5m $ and using the condition $ h(3) = 10 $, we successfully determined that:\n$$\nm = \frac{13}{5}\n$$", "This solution ensures the function accurately models the given data point, demonstrating a fundamental technique in function analysis—evaluating expressions with unknown parameters and solving for them based on known values.", "---", "Keywords: $ h(t) = t^2 - 4t + 5m $, function evaluation, solve for $ m $, algebra problem, $ h(3) = 10 $, quadratic function, parameter determination", "Meta Description: Learn how to find the value of $ m $ in $ h(t) = t^2 - 4t + 5m $ using substitution with $ t = 3 $ and $ h(3) = 10 $. Step-by-step explanation for solving functions."]
