The Lions Field

The Lion’s Field

Ajani's Notebook:

Ajani only knew a few pieces of information which he had to use to solve how far he needed to walk to reach Eagle Rock. He could only go west or east because crossing the field would be too dangerous for the lamb. He also needed to know how fast he had to travel to reach Eagle Rock in two days time.

To organize what he knew Ajani made a list with a corresponding drawing:

The approximate placement of Eagle Rock was NE of his village

Ajani’s List: The Eastern PathAjani’s List: The Eastern Path


Eagle Rock



To take the eastern route he needed to travel east 9km


Eagle Rock




Eagle Rock

9km



Turn 70° N of E


Eagle Rock

9km



Turn 70° N of E


70°

9km

Eagle Rock



Turn 70° N of E

To reach Eagle Rock travel West

70°

9km

Ajani’s List: The Eastern Ajani’s List: The Eastern PathPath

Eagle Rock

70°

9km

Ajani’s List: Other KnownsAjani’s List: Other Knowns

Eagle Rock

To cross the field diagonally NE and intersect the 70° N of E path it was a 20km journey

70°

9km

20km

Ajani’s List: Other KnownsAjani’s List: Other Knowns

Eagle Rock

To cross the field diagonally NE and intersect the 70° N of E path it was a 20km journey

Step 1: Label the diagram

70°

9km

20km

Eagle Rock

Solving the TriangleSolving the Triangle


70°

9km

20km

Eagle Rock


a

b

c

AB

C


Step 2: Solve for Angle A

70°

9km

20km

Eagle Rock


a

b

c

AB

C

Something to Think About;Something to Think About;

The total degree measurement of a straight line is 180°


180°

The total degree measurement of a straight line is 180°



Since a straight line is equal to 180 °

70°

9km

20km

Eagle Rock


a

b

c

AB

C




Angle A = 180 ° - 70 ° Angle A = 110 °

70°

9km

20km

Eagle Rock


a

b

c

AB

C




Angle A = 180 ° - 70 ° Angle A = 110 °

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

There are two equations that may be used to solve for non-right angle triangles:




The Sine Law Equation:

•Takes the Sine of an angle and divides it by the corresponding angles side

•Is used when three pieces of information is known

•Is not compatible with Side-Angle-Side or Side-Side-Side triangles



•Is with Side-Angle-Side or Side-Side-Side triangles

•Isn’t recommended because tedious work

The Cosine Law Equation:



Step 3: Solve for Angle C

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °


Side c is known therefore Angle C can be solved for with the Sine Equation

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

To solve for angle C cross multiply:



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °


And divide by 20 to isolate Sin C:



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °



Solve and take the inverse Sin to calculate the angle.



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °



Solve and take the inverse Sin to calculate the angle.

25.02°




Step 4: Solve for Angle B

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

The sum of all three angles in a triangle is equal to 180°


A

B C

Therefore: A+B+C=180°


Angle B can be solved for by subtracting Angles A and C from 180°

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°


Angle B can be solved for by subtracting Angles A and C from 180°

Angle B = 180° - (110° + 25.02°) = 44.98°

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

44.98°





Step 5: Solve for Side b

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

44.98°


Now that angle B is known side b can be calculated using the Sine Equation.

70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

44.98°

It is always best to use information originally given in a question to solve for

other unknowns


•There has not been previous rounding

•More accuracy



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

44.98°



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

Cross multiple and divide to isolate b:

44.98°



70°

9km

20km

Eagle Rock


a

b

c

AB

C

110 °

25.02°

Cross multiple and divide to isolate b:

15.04 km

44.98°

Ajani’s List: Other KnownsAjani’s List: Other Knowns•Eagle Rock is 72° N of E from the village

70°

9km

20km

Eagle Rock

a

b

c

AB

C

110 °

25.02°

15.04 km

44.98°

Ajani’s List: Other KnownsAjani’s List: Other Knowns•Eagle Rock is 72° N of E from the village

70°

9km

20km

Eagle Rock

a

b

c

AB

C

110 °

25.02°

15.04 km

72°

44.98°

70°

9km

20km

Eagle Rock

a

b

c

AB

C

110 °

25.02°

15.04 km

72°

44.98°






Step 6: Re-label new triangle


70°

9km

20km

Eagle Rock

a

b

c

AB

C

110 °

25.02°

15.04 km

72°

44.98°


Since both triangles share:

•A Common Side : a

•Common Angles: C and B


70°

9km

20km

Eagle Rock

a

b

c

AB

C

110 °

25.02°

15.04 km

72°

44.98°





The rest of the triangle can be labeled based on these three pieces of information.


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

110 °

25.02°

15.04 km

72°

44.98°





The rest of the triangle can be labeled based on these three pieces of information.


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

110 °

25.02°

15.04 km

72°

44.98°









70°

9km

20km

Eagle Rock

a

b

c

A

B

C

110 °

25.02°

15.04 km

72°

44.98°


Both the 72° and 44.98 ° angles begin on the same axis then:


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

110 °

25.02°

15.04 km

72°

44.98°



Angle B= 72° - 44.98°


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

110 °

25.02°

15.04 km

72°

44.98°



Angle B= 72° - 44.98°

Angle B= 27.02°


27.02 °

When a transversal line passes through two parallel lines corresponding angles are formed.


When a transversal line passes through two parallel lines corresponding angles are formed.

This means that the angles on alternate sides of the transversal are the same.











70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°

44.98°

27.02 °

15.04 km


Side a is a transversal line


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°

44.98°

27.02 °

15.04 km



This means that Angle C is congruent to 44.98°


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°

44.98°

27.02 °

15.04 km



This means that Angle C is congruent to 44.98° this is because it is on the alternate side of the transversal line to the original 44.98°.


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°

44.98°

27.02 °

15.04 km

44.98°











70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°

44.98°

27.02 °

15.04 km


Since two angles are known:


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

44.98°



Angle A = 180 ° - (44.98° + 27.02°)


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

44.98°



Angle A = 180 ° - (44.98° + 27.02°)

Angle A = 108°


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

44.98°108°












70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

108° 44.98°


The Sine equation can be used to solve side b:


70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

108° 44.98°




70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

108°

Cross multiply and divide to isolate b

44.98°




70°

9km

20km

Eagle Rock

a

b

c

A

B

C

72°27.02 °

15.04 km

108°


44.98°

9.55km











Step 11: Solve the Eastern Route


70°

9km

20km

ac

A

B

C

72°27.02 °

15.04 km

108° 44.98°

Eagle Rockb 9.55km


To Solve for the entire distance of the eastern route:


70°

9km

20km

ac

A

B

C

72°27.02 °

15.04 km

108° 44.98°

Eagle Rockb 9.55km



Eastern Route= 9km+15.04km+9.55km


70°

9km

20km

ac

A

B

C

72°27.02 °

15.04 km

108° 44.98°

Eagle Rockb 9.55km



Eastern Route= 9km+15.04km+9.55km

Eastern Route= 33.59km


70°

9km

20km

ac

A

B

C

72°27.02 °

15.04 km

108° 44.98°

Eagle Rockb 9.55km

•The path to the west was 50° N of W.

70°

20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

Ajani’s List: The Western Ajani’s List: The Western PathPath

•The path to the west was 50° N of W.

70°

20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb


50°

•The path to the west was 50° N of W.•Eagle Rock is directly East from the first path

20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb


50°

Step 1: Solve Side c

20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

50°



ac

A

B

Cb

When working with multiple triangles:

•the side of a previous triangle


ac

A

B

Cb

When working with multiple triangles:

•the side of a previous triangle

•Can be used as the side of another triangle


20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

50°


Side c can be solved using the Sine Formula and values from the previous triangle:


20km

ac

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

50°



Cross multiply and divide to isolate c


20kma

c

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

50°



Cross multiply and divide to isolate c

14.86km

c= 14.864729… = 14.86km

Step 1: Solve Side cStep2: Solve Angle A

20km

c

A

B

72°27.02 °

108° 44.98°

Eagle Rock

50°


14.86km

a

C b

Step2: Solve Angle A

Angle A is on the same horizontal line as the Angle of 108°

20km

c

A

B

72°27.02 °

108° 44.98°

Eagle Rock

50°


14.86km

a

C b



Therefore Angle A = 180 °-108 °

20km

c

A

B

72°27.02 °

108° 44.98°

Eagle Rock

50°


14.86km

a

C b



Therefore Angle A = 180 °-108 °Angle A = 72°

20kma

c

A

B

C

72°27.02 °

108° 44.98°

Eagle Rockb

50°


14.86km

72°

Step 1: Solve Side cStep 2: Solve Angle AStep 3: Solve Angle B

20km

c

A

B

72°27.02 °

108° 44.98°

Eagle Rock

50°


14.86km

a

C b

72°

Step 3: Solve Angle B

Since the angles of 50° and 72° are both located on the x axis subtracting these from 180° will give the value of Angle B.

20km

c

A

B

72°

108° 44.98°

Eagle Rock

50°


14.86km

a

C b

72°



Angle B= 180° - (50°+72°)20km

c

A

B

72°

108° 44.98°

Eagle Rock

50°


14.86km

a

C b

72°



Angle B= 180° - (50°+72°)Angle B= 58° 20km

a

c

A

B

C

72°

108° 44.98°

Eagle Rockb

50°


14.86km

58°

72°

Step 1: Solve Side cStep 2: Solve Angle AStep 3: Solve Angle BStep 4: Solve Angle C

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

Step 4: Solve Angle C

Since we know a triangle is equal to 180°

Angle C= 180° - (58°+72°)

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

72°

58°



Angle C= 180° - (58°+72°)Angle C= 50°

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

72°

58°

50°



Angle C= 180° - (58°+72°)Angle C= 50°

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

72°

58°

50°

**Note to check if your angles are correct:

Angle B plus 50° should equal 108° because it is on the alternate side of the transversal c.

50 ° + 58 ° = 108 °

108°

Step 1: Solve Side cStep 2: Solve Angle AStep 3: Solve Angle BStep 4: Solve Angle CStep 5: Solve side a

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

Step 5: Solve side a

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°


20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

Cross multiply and divide to isolate a


20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

Cross multiply and divide to isolate a

a= 18.448929…

= 18.45km

18.45km

Step 1: Solve Side cStep 2: Solve Angle AStep 3: Solve Angle BStep 4: Solve Angle CStep 5: Solve Side b

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

18.45km

Step 5: Solve Side b

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

18.45km


20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

Cross multiply and divide to isolate b 18.45k

m


20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°


b= 16.450735…

= 16.45km

16.45km

18.45km

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb

Solving for the Western Solving for the Western RouteRoute

14.86km

58°

72°

16.45km

18.45km

To solve for the Western Route from the village to Eagle Rock:

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

16.45km

18.45km


Add the two paths:

Western Route= 18.45+16.45

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

16.45km

18.45km


Add the two paths:

Western Route= 18.45+16.45

Western Route= 34.90

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb

Choosing the Shortest RouteChoosing the Shortest Route

14.86km

58°

72°

16.45km

18.45km 15.04 km

9.55km

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

16.45km

18.45km 15.04 km

9.55kmRoute One = 9km + 15.04km + 9.55km

= 33.59km

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

16.45km

18.45km 15.04 km


= 33.59km

Route Two= 18.45km + 16.45km

= 34.90km

20kma

c

A

B

C

72°

108° 44.98°

Eagle Rockb


14.86km

58°

72°

16.45km

18.45km 15.04 km


= 33.59km

Route Two= 18.45km + 16.45km

= 34.90km

According to Ajani’s calculations route one would be the shortest path to Eagle Rock.

SpeedSpeedAjani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has walking to complete the journey how long will it take to travel the 33.59 km?

SpeedSpeedAjani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km?Step one: set up a ratio to use to calculate the speed

Since Ajani is going to be walking for 12 hours each day for two days you need to know how far he will be able to walk in 24 hours.

Cross multiply and divide to isolate the value of the km.

Km= 36km

Since Ajani only has to travel 33.59km he will be able to walk at a normal pace since he will be able to cover 36km in 24 hours.

SpeedSpeedAjani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km?

Step one: set up a ratio to use to calculate the speed




Km= 36km




Km= 36km

Since Ajani only has to travel 33.59km he will be able to walk at a normal pace since he will be able to cover 36km in 24 hours.

The Lions Field

Entertainment & Humor

Transcript of The Lions Field