Solution: We are given $ D(t) = kt^2 + 5t $ and that $ D(3) = 48 $. Substituting $ t = 3 $: - Malaeb
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:
📰 which of the following correctly describes a point 📰 what is rrna 📰 meaning bildungsroman 📰 Pdf To Brainrot 9729504 📰 Game Changing Cod Black Ops 6 Zombies Revealedcan You Survive The Ultimate Navy Seal Level Zombie War 2239947 📰 This Secret Loreal Spray Transforms Your Look Like Magic 885124 📰 Newest Samsung 7735144 📰 Cant Fight The Moonlight Leann Rimes 9504519 📰 Ssh Tunnel Mac 4839575 📰 Step Into Virtual Style These Fashion Games Are Outselling Players Alive 3813820 📰 Kingdom Of Heaven Directors Cut Exposes The Real Battle Behind The Closed Ending 5242013 📰 The Epic Face Off Cartoon Wars You Need To Watch Before Its Gone 7525884 📰 Your Escape Awaits Firestone Destination Le3What They Never Wanted You To Find 9821250 📰 Diagram Of Er Model 7879025 📰 Who Sold Her Purrington Cat In The Ultimate 1998 Masterpiece 1731188 📰 Veins Ubiquiti Stock The Secret Boost Driving Tech Growth You Wont Want To Miss 7733542 📰 Alea Iacta Est 8519434 📰 Seneca Trattoria Secret Hidden Talent You Havent Tried Hour Yetwatch This 7615789Final 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.
