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Exploring Quadratic Modeling: Methods and Applications in Real-World Problems

This guide presents five effective methods for solving quadratic problems, including using RREF with three data points to find a quadratic equation, regression techniques for curve fitting, revenue analysis for maximizing profits in sales, area calculations for geometric models, and projectile motion modeling for height over time. Each method is illustrated with examples, such as calculating maximum revenue from unicycles, determining area with moats, and predicting the height of a thrown ball. Learn to apply these techniques to various mathematical and real-world scenarios.

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Exploring Quadratic Modeling: Methods and Applications in Real-World Problems

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  1. Modeling Problemswith Quadratics

  2. Method 1: RREF • Uses three data points, a system of equations, and matrices. • A quadratic equation can be found through any three points. Ex: (0,5) (2,3) (6,-29) -- ax²+bx+c=y 0a+0b+c=5 4a+2b+c=3 36a+6b+c=-29 Use RREF on the calculator to find a, b, and c. y=-2x²+8x-5

  3. Method 2: Regression • Use the data points to find a curve of best fit. Ex: Use Quad Reg to find a regression equation. y=0.009x²-2.104x+120.333

  4. 1. What is the water level at 25 seconds? (73.358 mm) 2. What is the water level at 1.5 minutes? (5.262 mm) 3. What is the water level at 3 minutes? (no solution – the water has already emptied)

  5. Method 3: Revenue Problems Revenue = price (# sold) Ex: The expression -2.5p+500 models the number of unicycles a company sells per month, where the price, p, can be set between $70 and $120. What is the maximum revenue? R=p(-2.5p+500)=-2.5p²+500p Max is at the vertex. P=-500/(2*-2.5) =-500/-5=100 R=-2.5(100) ²+500(100) = 25000 Maximum Revenue is $25,000 when unicycles are $100

  6. Method 4: Area Problems Ex: Mark wants to build a moat around his desk. His desk is 4 ft by 5 ft and the moat will be x ft wide. What equation models the total area of the desk and the moat? A=(4+2x)(5+2x) A=4x²+18x+20 ft² What will the total area be if the moat is 3 ft wide? (110 ft²) What is the area of the moat? (90 ft²)

  7. Method 5: Projectile Problems y=-16t² + vt+ h gives the model for height (in feet) of an object over time (in seconds) -16 is the gravity constant (use -4.9 for meters) v is the initial velocity h is the initial height

  8. Ex: A ball is thrown from two feet off the ground at 30 ft/sec. Find the model for the ball’s height over time. y=-16t²+30t+2 Find the maximum height. 16.063 feet When is the ball 12 feet high? Set the equation equal to 12 and solve for t using Quad Formula. When does the ball hit the ground? 1.939 sec

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