Problem Solving Using Polynomial Equations

1. Problem Solving Using Polynomial Equations Objective: To solve problems using polynomial equations

2. Physics • The motion of an object with respect to gravity can be modeled by a quadratic equation • When an object is launched upward with an initial velocity v, its height h from the launch point after t seconds is • h = vt – 4.9t2 metersh = vt – 16t2 feet

3. Example: • In celebration a person fires a bullet into the air. The bullet shoots upward at a speed of 350 m/s. • A) How long will it be till the person who fired the gun may accidentally shoot himself in the head. • B) What is the maximum height the bullet will reach before it comes speeding back to earthThis problem was inspired by the hundreds of Iraqi’s who managed to injure themselves from shooting guns in the air during celebration of various events

4. Solution: • A) The height at which the bullet will strike the ground is 0 • The velocity is 350 m/s • h = vt – 4.9t2 • 0 = 350t – 4.9t2 • 0 = t(350 – 4.9t) • t = 0 or 350 – 4.9t = 0 • t = 0 or t ≈ 71.429 s

5. B) What is the maximum height the bullet will reach before it comes speeding back to earth • The bullet should reach its maximum height at the halfway point t = 71.429/2 t = 35.7145 • h = vt – 4.9t2 • h = 350(35.7145) – 4.9(35.7145)2 • h = 6250 m • Note: these types of problems do not take into consideration, friction, or wind resistance and so can sometimes be unrealistic.

6. Area • A tennis court is twice as long as it is wide. There is also an extra 12 feet of space outside of the court lines. If the tarp covering the field is 6426 square feet what are the dimensions of the court?

7. ↕12ft 12ft ↔ 12ft ↔ ↕12ft A = lw • A = 6426 ft • l = 2w • A = lw • 6426 = (2w)(w) • Add on the border • 6426 = (2w + 12 + 12)(w + 12 + 12) • 6426 = (2w + 24)(w + 24)

8. 6426 = (2w + 24)(w + 24) • 6426 = w(2w + 24) + 24(2w + 24) • 6426 = 2w2 + 24w + 48w + 576 • 6426 = 2w2 + 72w + 576 • 6426 – 6426 = 2w2 + 72w + 576 – 6426 • 2w2 + 72w – 5850 = 0 • 2(w2 + 36w – 2925) = 0 • 2925 = 32• 52 • 13

9. Almost Done • 2(w2 + 36w – 2925) = 0 • 75 • -39 = -2925 75 + -39 = 36 • w2 +75w – 39w – 2925 = 0 • w(w + 75) – 39(w + 75) = 0 • (w – 39)(w + 75) = 0 • w – 39 = 0 or w + 75 = 0 • w = 39 or w = -75

10. Solution • w = -75 is discarded because it doesn’t make sense in real life. w = 39 • l = 2w l = 2 • 39 = 78 • The dimensions of the court are 39 ft. by 78 ft.