How do you find out how high an arrow might fly?

For exampleIf an arrow is shot at 50 m/s

upwards, when will it be above a certain height?

In this lesson you will learn how to create and solve

inequalitiesby using a quadratic

relationship

Let’s Review

x2 - x -12 ≤ 0

-5? 5?0?-3 ≤ x ≤ 4

Quadratic inequalities have a range of values that make the inequality statement true.

A Common Mistake

Not verifying answers after solving.

Core Lesson

We will investigate the following problem: An arrow is shot straight upwards; its height

(in meters) above the ground can be modeled with an equation: h(t) = -10t2 + 50t

During what time(s) will the arrow be above 40 meters above the ground?

Core Lesson

h(t) = -10t2 + 50t40 ≤ -10t2 + 50t40 = -10t2 + 50t

0 = -t2 + 5t - 4(-t+4)(t-1) = 0t= 1, 4 0? 5?3?

1 ≤ t ≤ 4

Core Lesson

40 ≤ -10t2 + 50t

1 ≤ t ≤ 4

VERIFY: what do my answers mean? do they make sense?

In this lesson you have learned how to create and solve quadratic inequalitiesby using a quadratic

relationship

Guided Practice

We will investigate the following problem: An arrow is shot straight upwards; its height

(in meters) above the ground can be modeled with an equation: h(t) = -10t2 + 50t

During what time(s) will the arrow be below 60 meters above the ground?

Guided Practice

h(t) = -10t2 + 50t

60 = -10t2 + 50t60 ≥ -10t2 + 50t

0 = -t2 + 5t - 6(-t+2)(t-3) = 0t= 2, 3 1? 4?2.5?

t ≤ 2, t≥3

Guided Practice

60 ≥ -10t2 + 50t

VERIFY: what do my answers mean? do they make sense?

t ≤ 2, t≥30 ≤ t ≤ 2, 3 ≤ t ≤ 5

Extension Activities

a. Use this equation to investigate how far objects fall when dropped, according to time: h(t)= -10t2

b. Go back to the practice problems and find the maximum height of the arrow. You can also investigate at what time it returns to the ground, etc

Quick Quiz1. A flare is launched from a stranded boat to call for help; the height of the flare is modeled by the equation: h(t)=-10t2 + 100t When will the flare be above 160 m, so the coast guard can see it?2. Angela opened up a new restaurant, and predicted that its profit could be modeled with the equation: p(t) = 1125(t-1)2 – 10,125 During what time spans will Angela’s restaurant NOT make a profit?