Solution: We are given $ D(t) = kt^2 + 5t $ and that $ D(3) = 48 $. Substituting $ t = 3 $: - Richter Guitar
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:
📰 tn department of health 📰 tennesee 📰 mark chesnutt hospitalized 📰 You Wont Guess How Pirates 3 Outdoes Its Predecessorsheres The Shocking Twist 5543156 📰 Way Too Early 662634 📰 How Jeff The Shark Became Internet Sensationzoom In On The Shocking Moment 1139616 📰 Git Reset Hard 5005877 📰 The Equation Is X X 2 146 8420953 📰 Preppy Christmas Wallpaper Thatll Put A Sharp Twist On Your Holiday Vibes 616850 📰 Mcdonalds Monopoly Game Winning Pieces 1998803 📰 Hubspot Share 509239 📰 Action Pack Microsoft 8863888 📰 Space Wars Battleground Revealedhidden Alien Tactics Will Shock You You Wont Believe Whats Coming Next 3225543 📰 A Car Accelerates From Rest At A Constant Rate Of 3 Textms2 How Far Does It Travel In 10 Seconds 3847307 📰 How The Usd To Xof Exchange Rate Just Made African Smart Investors Rich 4605131 📰 Unlock The Power Of The 2000 Jeep Grand Cherokeeyoull Want One Today 9728149 📰 Wyatt Kelce 6178957 📰 This Simple Length Changes Everythingdiscover The Hidden Power Of 45 Inches In Feet 7685326Final 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.
