Distance Calculations

Distance The Port of Long Beach to China

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Objective and Standard •

Students will solve real world cost problems by using the distance formula.

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California Standards: Geometry 15: Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles. Geometry 17: Students prove theorems using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.

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The Cartesian Coordinate System

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The Cartesian coordinate system was originally developed by the French philosopher and mathematician René Descartes. You may recognize his famous quote: “ I think, therefore, I am.” 3

The Cartesian Coordinate Plane

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Legend has it that Descartes first devised the coordinate system while watching a bug crawl across the ceiling.

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His development of the coordinate plane helped to bring together the two disciplines of Algebra and Geometry.

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Purpose •

The coordinate system allows us to study the distance between two points on a plane.

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We can use this information to solve real world problems such as calculating costs associated with shipping cargo to and from the Port of Long Beach. 5

Review of the Basics •

The Cartesian coordinate plane has four quadrants. The quadrant the point occupies is dependent upon the x and y coordinates (x, y) (or vice – versa).

Y

Quadrant II (- ,+)

Quadrant I (+,+) X

Quadrant III (-,-)

Quadrant IV (+,-)

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Plotting Points •

We plot points according to the ordered pair, (x, y). Let’s plot point (5,4).

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•

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The x-coordinate tells us how far from the origin to move, left or right.

Y 6

(5,4)

4 2

2

4

6

X

The y-coordinate tells us how far to move from the origin, up or down.

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Practice •

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Draw a coordinate plane. Label the x and y axis. Plot the following points. A. (-3, -3) B. (4,2) When you are done, compare your work with your neighbor.

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Check your answer Y

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Draw a coordinate plane. Label the x and y axis. Plot the following points. A. (-3, -3) B. (4, 2)

6 4

B (4, 2) 2

2

4

6

X

A (-3,-3)

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Locating the Quadrant Again, the quadrant the point is in will depend upon the signs of x and y. For example, the point (4, -3) would be located in quadrant IV.

Y

II

I

6 4 2

2

III

4

6

X

VI

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Shout Out What quadrant is the point in? Y

(-2,3)? (Quadrant II) C. (-4, 0) (B is on the X – axis) C. (2, -4) (Quadrant IV) A.

Quadrant II

Quadrant I

A

B

Quadrant III

C Quadrant IV

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Practice •

Plot the following points and identify the quadrant in which the they are located. 1. (5,-4) 2. (0,-3) 3. (2,1) 4. (-4,-5)

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Check your work Y

1.(5,-4) 2.(0,-3) 3.(2,1) 4.(-4,-5)

II

I

6 4 2

(2,1)

(0,-3) 2

4

6

(-4,-5) III

X

(5,-4) VI

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Pythagorean Theorem •

We will see that the distance formula is the Pythagorean theorem.

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If a2 + b2 = c2 where a and b are the lengths of the legs and c is the length of the hypotenuse and given p1 (x1,y1) and p2 (x2,y2),

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Then a is the difference between x1 and x2 and b is the difference between y1 and y2, and c is the length between the two points (the distance). 14

Distance Formula and the Pythagorean Theorem

c2 = a2 + b2

Y

or

(xx2, y22)

2

Distance -

( x2 − x1 ) 2 + ( y2 − y1 ) 2 -

(xx1, y1)

( x2 − x1 ) 2 + ( y2 − y1 ) 2

x

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An Example •

Find the distance between the points (3, 2) and (-4, 6).

(-4,6) 6 4

(3,2)

2

-4

-2

2

4

6

X

D =  + 

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Check for Understanding •

Find the distance between the points (1, 5) and (4,2).

6

(1,5)

4

(4,2)

2

-4

-2

2

4

6

X

• Raise your right hand if your answer is 3√2, raise your left hand if it is not. 17

Calculate the distance between the given points. 1. (-4,3) and (7,-5)

2. (3,8) and (10,-12)

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The Port of Long Beach •

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The Port of Long Beach is the second busiest port in the United States and the 12th busiest port in the world. By far, the largest inbound trading partner is China accounting for 41% of the containers coming into the port and 42% of the cargo leaving the port. 19 Input

The Port of Long Beach

The Port of Long Beach covers 11.9 square miles and 35 miles of waterfront.

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The Port of Long Beach It has 10 piers…

21 Input

The Port of Long Beach …80 Berths…

Pier T with 5 Vessels at Berth

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The Port of Long Beach …and 71 post-panamax gantry cranes.

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The Port of Long Beach

Each year more than $100 billion worth of cargo passes through the Port of Long Beach.

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If cargo has come from China, it has covered a vast distance…

United States Long Beach

China

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…on a massive container ship.

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Applying the Distance Formula •

We will use the distance formula to calculate the distance (on a two dimensional map) that the container ships must travel from China to Long Beach. We will then calculate the approximate fuel cost of these voyages.

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Since we are using a coordinate plane, these calculations will be approximations as the distance formula is a planar tool…

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Planar Representation of Earth

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Applying the Distance Formula

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We will use the distance formula to calculate the distance (on a two dimensional map) that the container ships must travel from China to Long Beach. We will then calculate the approximate cost of these voyages.

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Since we are using a coordinate plane, these calculations will be approximations as the distance formula is a planar tool, and the earth itself is round. 29

Planar Representation of Earth

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The Hanjin Amsterdam The Hanjin Amsterdam has just finished unloading and loading cargo at the Port of Shanghai (point S).

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The Hanjin Amsterdam The Hanjin Amsterdam has just finished unloading and loading cargo at the Port of Shanghai (point S). It is setting sail for the Port of Long Beach.

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Shanghai, China

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The Hanjin Amsterdam The Hanjin Amsterdam has just finished unloading and loading cargo at the Port of Shanghai (point S). It is setting sail for the port of Long Beach. Using the data on the figure, calculate the distance the Hanjin Amsterdam must sail to reach the Port of Long Beach (point L). 35

Y

(3490.8, 2331.5)

L

S (-3414.8, 2152)

X

Solution

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D = x2 − x1 + ( y 2 − y1 ) 2 D = (−3414.8 − 3490.8) 2 + (2152 − 2331.5) 2 D = (−6905.6) 2 + (−179.5) 2 D = 47,687,311.36 + 32,220.25 D = 47,719,531.61 D = 6907.9mi

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Check for Understanding •

The OOCL Los Angeles is sailing from Zhanjiang to Honolulu and then on to Long Beach. (Take a moment to copy this information on your paper.) Using the information on the following graph, calculate the distance the ship will traverse. Write your answer on your white board. 38

(3490.8, 2231.5) Long Beach

Zhanjiang (-2465, 2234.7)

Honolulu (1423.5, 1470.7)

Solution Distance from Zhanziang to Honolulu 2

D = x2 − x1 + ( y 2 − y1 ) 2 D = 1,423.5 − (−2,465) 2 + (1,470.7 − 2,234.7) 2 D = 3,888.52 + (−764) 2 D = 15,120,432.25 + 583,696 D = 15,704,128.25 D = 3,962.8mi

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(3490.8, 2231.5) Long Beach

Zhanjiang (-2465, 2234.7)

Honolulu (1423.5, 1470.7)

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Solution Distance from Honolulu to Long Beach 2

D = x2 − x1 + ( y 2 − y1 ) 2 D = 3490.76 − 1423.5 2 + (2231.5 − 1470.7) 2 D = 2067.26 2 + (760.8) 2 D = 4,273,563.9 + 578,816.64 D = 4,852,380.5 D = 2202.8mi

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Solution Continued… • • •

Total Distance: 3962.8 + 2202.8=6165.6 mi The total distance the ship traveled is approximately 6,165.6 mi.

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How much does fuel cost? •

Using data from 2005 we can estimate the cost per mile for shipping between China and Long Beach as $87.50 per mile.

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Use this information and your answer from the last problem to calculate the total cost of fuel for the OOCL Los Angeles to sail from Zhanjiang to Honolulu and then on to Long Beach.

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Cost of Fuel • • • •

Total Distance: 6165.6 mi. Approximate cost of fuel per mile: $87.50 (6165.6)(87.5)=$539,490 The approximate cost of fuel for the trip from Zhanjiang, China to Honolulu, HI and then to Long Beach, CA was $539,490. 45

Check for Understanding •

Turn to your neighbor and explain to them how you calculated the fuel cost for the OOCL Los Angeles.

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Practice In 2006, fuel costs had risen to approximately $171.86 per mile. Calculate the total distance the OOCL Los Angeles will travel if it departs out of Shanton, docks in Long Beach and then returns to Shanton. Calculate the approximate fuel cost of the roundtrip voyage.

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(3490.76, 2231.5) Long Beach

Shanton (-4009.2,1545.5)

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Solution 2

D = x2 − x1 + ( y 2 − y1 ) 2 D = 3490.8 − (−4009.2) 2 + (2231.5 − 1545.5) 2 D = 7500 2 + (686) 2 D = 56250000 + 470596 D = 56720596 D = 7531.3mi

Round Trip: (7,531)(2)= 15062.6mi Cost: 171.86(15062.6)= $2,588,661

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Solution •

If the one way distance was 7,531mi, then the round trip voyage was approximately 15,062.6mi.

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Therefore the cost of the total voyage was approximately, $2,588,66.

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Closure •

Write a summary of today’s lesson, focusing on the real-world application of the distance formula. Looking at the answers for the last two problems, how can shipping costs affect consumers and producers in the United States?

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