Boat across the RiverTerm 2 Advanced Math Project

Done By

Essa Essa Ahmed Ali Salem bin hadier Abdulla Abbas Ahmed obiad

Section : 11-13

Lets start

• Addition of vectors• Pictorially vector addition is pretty

straightforward. If you want to add the

vectors A and B, you just take the second vector, represented by an arrow, and translate it so that it's tail is at the head of

the first vector. A + B is just a vector that starts at the tail of the first arrow and goes to the head of the second, so the whole

thing, vectors A , B , and A + B , form a triangle, as shown:

• Task 1: Students will search the web to find real life examples on working with vectors, e.g. adding forces, projectile, components, vectors in 3D, etc…

Now the nice thing about this definition is that it's easily seen that A + B = B+ A , and also the rule for adding in terms of components couldn't be simpler. Look at the x components for example. What is the x component of the vector A + B ?. Its just the addition of the x components of the two vectors, AX + BX . The following picture easily sees this. We just show ``project'' the components of A, B, and A+B down close to the x - axis. You see there's a of length AX , one of length BX , and one which is the x component of AX + BX which I labeled ( AX + BX )2 . You see from that picture that the total length of the red and green lines, is the length of the blue line. That's what we wanted to show. The same holds true in the y direction, so that the y component of A + B is just AY+ BY .

So vectors add nicely in terms of their x and y components. It's much messier in terms of the vectors polar representation.

• Task 2: 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.

10 kM

5 kM

11.

1 kM

•Determine

Using Pythagorean thermo :- R=√102+52

R = 11.1 km

Find the angle which is “direction” :-

As we know we us θ=tan-1(opp÷adj)θ=tan-1 (510)=26.5° but relative to x-axis it is 90-26.5 = 63.5°

determine the magnitude

determine the diriction

•Task 3

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.

A= 300 N at 200° Ax=300cos200=-281.9Ay=300sin200=-102.60

B= 500 N at 150° Bx =500cos150=-433.01By =500sin150=250

Rx=Ax + Bx= -433.01 +(-281.9)= -814.91 N Ry=Ay+By= 250+(-102.6)= 147.4 N R=Rx2+Ry2

=(-814.91)2+(147.4)2

= 828.13 N

• Find the magnitude of the resultant force

• 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 15Dhs for each small package, and 15 for each large package.

•Task 4

a. Write an inequality to represent the weight of the packages in kg the boat can hold. Let x= Small box, Let y= large box

20 x + 25 y≤ 1200

• b. Write an inequality to represent the volume in cubic meter of packages the boat can hold. 1200x+1600y ≤ 7.200000

a. Write an inequality to represent the weight of the packages in kg the boat can hold.

Let x= Small box, Let y= large box 20 x + 25 y ≤ 1200

C. Graph the system of inequality

The Convex set is ( 0 , 0 ) , ( 0 , 48 ) , ( 60 , 0 )

d. Write a function that represents the amount of money the delivery service will make on each boat hold.

F(X) = 15 x + 20 y

e. Find the number of each type of the boxes that should be placed on a boat to maximize revenue. F(X) = 15 x + 20 yConvex set is ( 0 , 0 ) , ( 0 , 48 ) , ( 60 , 0 ) f(0,48) = 15(0) + 15(48) = 720$ f(0,0) = 15(0) + 15(0) = 0f(60,0) = 15(60) + 15(0) = 900$ they should deliver 60 small package and 0 large package.

f. What is the maximum revenue per boat? The maximum revenue if 900$ is it shown in part “e”.

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?

The magnitude is: - √452+(24)2+(110)2=121.24

• 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

•Task 5

W=F.d SIN Θ W= 200 X 6 SIN 30= 600 J