Solution: We are given $ D(t) = kt^2 + 5t $ and that $ D(3) = 48 $. Substituting $ t = 3 $: - inBeat
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:
📰 Unblocked Crazy Games Thatll Steal Your Brain—Exploding with Rewards and Brainrot! 📰 Crazy Brain-Robbing Fun—Unlock These Free Games That Keep You Hooked Forever! 📰 Steal a Brainrot? These Unblocked Crazy Games Will Ruin Your Brain (For Good!) 📰 Final Destnation 5 5857570 📰 Discover The Shocking Trick To Print Outlook Emails Instantly No More Scrolling 2014786 📰 Doha Airport Hotel 2393293 📰 Dating Sim Simulation Games 128428 📰 Solo Roth 401K 5305696 📰 Ghana Black Stars Roster 3521721 📰 Geneva And 9573312 📰 Free Games Download Website 8202052 📰 City Of Imperial Ne 3911825 📰 2019 World Series 2663472 📰 Discover The Secret Behind Mrge Stocks Blazing Riseinvest Like A Pro 6387232 📰 Mets News Alonso 1581170 📰 Cast Of Santa Clarita Diet 5380683 📰 Why Everyones Obsessed With Fionna Adventure Time Must Watch Now 1705314 📰 Petra By Night Touches Thousandsuncover The Invisible Crowd 549301Final 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.
