Solution: We are given $ D(t) = kt^2 + 5t $ and that $ D(3) = 48 $. Substituting $ t = 3 $: - NBX Soluciones
Solving for the Constant in the Quadratic Function: A Step-by-Step Guide Using $ D(t) = kt^2 + 5t $
Solving for the Constant in the Quadratic Function: A Step-by-Step Guide Using $ D(t) = kt^2 + 5t $
When working with mathematical modeling in science, engineering, or economics, analyzing functions of the form $ D(t) = kt^2 + 5t $ is common. These quadratic equations often represent real-world phenomena such as displacement, revenue, or cost over time. In this article, we’ll walk through how to determine the unknown constant $ k $ using the known value $ D(3) = 48 $, offering a clear, step-by-step solution that highlights key algebraic techniques useful for educators, students, and professionals.
Understanding the Context
Step 1: Understand the Function and Given Information
We are given a quadratic function:
$$
D(t) = kt^2 + 5t
$$
This function models a quantity $ D $ that depends on time $ t $, with the unknown constant $ k $ needing to be determined. We’re also told that at $ t = 3 $, $ D(3) = 48 $.
Substituting $ t = 3 $ into the function gives:
$$
D(3) = k(3)^2 + 5(3)
$$
Image Gallery
Key Insights
Step 2: Substitute and Simplify
Now compute each term:
$$
D(3) = k(9) + 15 = 9k + 15
$$
Since $ D(3) = 48 $, equate the expressions:
$$
9k + 15 = 48
$$
Step 3: Solve for $ k $
Subtract 15 from both sides:
$$
9k = 48 - 15 = 33
$$
Now divide both sides by 9:
$$
k = rac{33}{9} = rac{11}{3}
$$
🔗 Related Articles You Might Like:
📰 Roblox Library Decals 📰 Roblox Exucuter 📰 Simon Says on Roblox 📰 Gold Testing Near Me 9960825 📰 This Moment Less Than A Second Altered Your Destiny Forever 7823509 📰 Nrsv Bible You Thought You Trusted Holds Shocking Truths Inside 5654551 📰 Is The Carlyle Group Lp Stock Your Next Big Gamble Shocking Market Move Sparks Lucrative Opportunities 2442043 📰 Unleash Warstrike Power Ready The Deadliest War Tactics Now 4856419 📰 Stanley Browns Secret Safety Plan Shocks Every Survival Expertheres How To Use It 3534337 📰 Whos Eligible The Shocking Truth About Medicaid For Everyone 986575 📰 Gallantly 5504078 📰 Activate Your God Hand Unlock Superhuman Powers With This Ancient Mystery 4501141 📰 Kingdom Hearts 3 4657399 📰 Pocket Sync 8569926 📰 You Wont Believe What Happened In Bondres The Facts Are Unmissable 626156 📰 Colorado State Bird 2031058 📰 Porcentaje Con Resultados Positivos 15 9234315 📰 Deer White House Loyalty Score Companies Proving Their Allegiance Today 1418795Final Thoughts
Step 4: Final Verification
To confirm, plug $ k = rac{11}{3} $ back into the original equation:
$$
D(t) = rac{11}{3}t^2 + 5t
$$
Now compute $ D(3) $:
$$
D(3) = rac{11}{3}(9) + 5(3) = 33 + 15 = 48
$$
The result matches the given value, verifying our solution.
Why This Technique Matters
This example illustrates a standard algebraic method for solving for unknown coefficients in quadratic functions—substitution followed by isolation of the unknown variable. Such skills are essential in fields ranging from physics (modeling motion) to finance (forecasting growth), where precise parameter estimation ensures accurate predictions.
Key Takeaways:
- Always substitute known values into the functional equation.
- Simplify expressions algebraically before isolating the unknown.
- Verify your solution by plugging it back into the original equation.
By mastering this step-by-step process, anyone can confidently solve similar problems involving quadratic models in real-world contexts.
