Mathematical Applications

1. Mathematical Applications Many of the mathematical skills learnt in this topic can be applied to everyday situations.

2. Dave, a school pupil, was planning to knife the head of discipline, Don. Dave keeps his arm straight while thrusting the blade and his whole arm pivots from his shoulder, creating a circular path with centre of his shoulder. From experience he knows that the most effective technique is to enter the stomach upwards with an angle of π/6 to the horizontal. Don’s stomach is at position (3,1). (a) Along what line must Dave’s shoulder exist if he is to knife Don accurately? (3, 1)

3. You can see that the shoulder must be along the line perpendicular to the path of the knife. π/6 2 2π/6 √3 m = -2/√3 y – b = m ( x – a ) y – 1 = -2/√3 ( x – 3 ) y = -2/√3 x + 2√3 + 1

4. Dave’s knife also passes through the point (2,1). (b) Where is Dave’s shoulder positioned? x = 2.5 Without equation it can be easily seen that Dave’s shoulder must be along a vertical line half way between the 2 points on the arc. Line x = 2.5 Shoulder exists where lines cross. y = -2/√3 x + 2√3 + 1 y = 1.58 (2.5,1.58)

5. In retaliation, Don launches a table at Dave. Dave is making a break for the door. Dave is travelling in a straight line with equation y = 5. Don throws a table with a parabolic path with equation y = -(x – 5)² + 15 (c) Assuming they cross paths at the same time, where will the table connect with Dave?

6. 1 The volume of blood in Don’s body is related to the graph of: Where 0 < x < ∞ e 0.2x - 2 When the volume of blood in his body drops below 3units Don will pass out. The x-axis is measured in minutes. (d) How long does Don have to find help?

7. Good luck in your new job, You’ll need it! Advanced Higher