 Advanced Math Project Done by: Yousef Mohammad Ali, Mahmood Ghanim, and Mohammad Shizawi, Ishaq Abdullah. Class: G Directed to: Mr. Yahiya Jamal Al Sakagi. Wiki’s address:

Task 1 T1: Students will search the web to find real life examples on working with vectors, e.g. adding forces, projectile, components, vectors in 3D, etc…  Most cultures use vector addition in very similar ways, and for very similar reasons. Either to solve for unknown distances, describe uniform motion, or create basic skeleton diagrams for 3-dimensional objects.  In modern society: students who are studying to become engineers use vector applications frequently as means of simplifying the visualization of 3-dimensional geometry, or calculating the area of a potential pyramid.  In past: vectors had various primary purposes depending on the culture it was used in. For example, in Ireland in the 1830's vectors were used as evidence to support the theory of complex numbers, and their position on the plane. Hamilton, a devoted Irish mathematician, used vectors to represent complex numbers in a two-dimensional space, and attempted to add a third and fourth vector in hopes of finding a location for both the complex and imaginary numbers.

Task 2

T2: If the boat is heading due north as it crosses a river with velocity of 10 km/h relative to water, the river has a uniform velocity of 5 km/h due to east. Graph the vectors in the chart below. Then determine the magnitude and direction of the boat’s velocity.  Solution: - R=√ R = 11.1 km θ= tan -1 ( 5/10) = 26.5 °

Task 2 T2: Two horses are pulling the boat along a canal, one horse exerts a force of 300 N at an angle20°, and the other exerts 500 N at angle 30°. Find the magnitude of the resultant force. -  With reference made to the positive x-axis (the boat representing the origin): A= 300 N at 200° Ax=300 cos200= N Ay=300 sin200= N

Task 2  B= 500 N at 150° Ry = Ay + By Bx =500 cos150= -433 N = By =500 sin150= 250 N = N  Rx=Ax + Bx R=√Rx 2 +Ry 2 = (-433) =√(-714.9)2+(147.4)2 = N = √ = N

Task 3 The boat is part of delivering company, it can hold 1200 kg and its capacity is 7.20 cubic meter. The service handles two types of boxes: small, which weight is up to 20 kg and no more than 1200cm3, and large which 25 kg each and 1600 cm3 each. The delivery service charges 10 Dhs for each small package, and 15 Dhs for each large package. (a) Write an inequality to represent the weight of the packages in kg the boat can hold. S= small boxes L= large boxes 20S+25L  1,200

Task 3 (b) Write an inequality to represent the volume in cubic meter of packages the boat can hold. (Converting 7.20 cubic meter to cubic centimeter = 7,200,000) 1,200S + 1,600L  7,200,000 Graph the system of inequality: - 20S+25L  1,2001,200S+ 1,600L = 7,200,000 Let x =0 let x=0 20(0)+25L=1,200 1,200(0) + 1,600L= 7,200,000 (0,48) (0,4500)

Task 3 Let y=0 let y=0 20S + 25(0)= S + 1,600(0)= (60,0) (6000,0)

Task 3 (c) Write a function that represents the amount of money the delivery service will make on each boat hold. F(x, y)= 10S +15L (d) Find the number of each type of the boxes that should be placed on a boat to maximize revenue. f(0,0)= 10(0)+15(0)=0 f(60,0)=10(60)+15(0)=600 f(0,48)=10(0)+15(48)= 720 (e) What is the maximum revenue per boat? It can deliver 0 small boxes and 48 large boxes per boat

Task 4 Suppose that the force acting on the boat can be expressed by vector 45,24,110 where each measure in the ordered triple represents the force in Newton. What is the magnitude of the force? =√ (45) 2 +(24) 2 +(110) 2 = N

Task 5 Workers are using a force of 200 Newton to push a cart up a ramp. The ramp is 6 feet long and is at 30° angle with the horizontal. How much work are the workers doing in the vertical direction? You may use sine ratio and formula W=F.d W= f d cos θ =200x6xcos(30) = J

The End…  Hopefully you liked the project and wish you a nice day…