Solution: We are given $ D(t) = kt^2 + 5t $ and that $ D(3) = 48 $. Substituting $ t = 3 $: - 500apps
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:
$$
D(3) = k(3)^2 + 5(3)
$$
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:
📰 The Verdansk Map Every Explorer Desperately Wants – Click to Reveal Its Hidden Treasures! 📰 "Verdansk Map Secrets Exposed: This Hidden Layout Could Change How You Play Forever! 📰 Imagine the Thrill: How the Verdansk Map Leads to Incredible Discoveries – Click to Explore! 📰 Magnesium Lotion That Drastically Changes Your Skin Overnight 📰 Magnesium Powder The Hidden Danger Lurking In Every Kitchen Corner 📰 Magnesium Powder Youre Not Supposed To Use In Cookingyoull Burn Your Kitchen Down 📰 Magnesium Threonate Secrets You Never Knew Could Transform Your Brain 📰 Magnetic Nail Enamel That Locks Your Glam Foreverfar Clearever 📰 Magnetic Tiles That Turn Ordinary Play Into Breakthrough Creativity 📰 Magneto Magnus Confronts Erikyou Wont Believe Whos Real Face Of Power 📰 Magneto Magnus Shocking Birth Eriks Untold Truth Revealed In This Clash 📰 Magneto Or Erik The Power That Shatters Worldsfinal Verdict Within Reach 📰 Magnets And Mystery The Magsafe Case That Acts Smarter Than You 📰 Magnitude Magic Cruelly Banned Official Rulings Shock Fans Forever 📰 Magnolia Flower Hides A Secret That Will Change How You See Beauty Forever 📰 Magnolia Road Holds A Power You Cant Imaginesee It Before Its Gone 📰 Magnolia Road Secrets No One Dares Mention Before Sunset 📰 Magnum Condoms You Never Dare To Admit Work Like A Prono More Surprises AgainFinal 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.
