Geometry Applications: Rectangular Coordinates

Geometry Applications: Rectangular Coordinates

[Music]

[Music]

ON SEPTEMBER 4, 1622 THE SPANISH GALLEON

NUESTRA SENORA DE ATOCHA SET SAIL FROM HAVANA, CUBA

BOUND FOR SPAIN.

IT WAS LOADED WITH TONS OF GOLD, SILVER, EMERALDS

AND ENOUGH TREASURE THAT WOULD ALLOW THE

SPANISH EMPIRE TO FUND ITS WARS IN EUROPE.

BUT THE ATOCHA HAD BEEN DELAYED IN ITS DEPARTURE

BY SIX WEEKS.

THIS PLACED THE DEPARTURE RIGHT IN THE MIDST OF THE

ANNUAL HURRICANE SEASON THAT TORMENTS THIS REGION.

YET ANY FURTHER DELAYS WERE OUT OF THE QUESTION.

SO THE ATOCHA DEPARTED

ACCOMPANIED BY 27 OTHER SHIPS.

AS THE SPANISH FEARED, THE FLOTILLA OF SHIPS

ENCOUNTERED A HURRICANE TWO DAYS INTO THE VOYAGE.

THE ATOCHA WAS PUSHED TOWARD THE REEFS

ALONG WHAT IS NOW THE FLORIDA KEYS.

SUSTAINING DAMAGE TO ITS HULL, COMBINED WITH THE

HEAVY WEIGHT OF ITS CARGO, THE ATOCHA SANK

OFF THE COAST OF THE DRY TORTUGAS ISLANDS.

ALL BUT A HANDFUL OF PEOPLE SURVIVED.

THE SPANISH SENT SHIPS TO TRY TO SALVAGE THE TREASURE

BUT THE TECHNOLOGY OF THE TIME MADE THIS DIFFICULT

AND THE SPANISH HAD LIMITED SUCCESS.

SO THE SPANISH GALLEON ATOCHA JOINED OTHER SHIPS

THAT OVER THE CENTURIES SANK ALONG THE WATERS

OFF THE FLORIDA KEYS.

IN 1969 A DETERMINED TREASURE HUNTER

BEGAN A QUEST THAT WOULD END 15 YEARS LATER

WITH THE RECOVERY OF MOST OF THE REMAINS OF THE SHIP.

DURING THAT TIME, BITS AND PCES OF THE TREASURE

WERE FOUND OVER A LARGE AREA OF THE SEA.

WHEN A WOODEN SHIP SINKS, THE WATER CAUSES THE WOOD

TO, OVER TIME, DECOMPOSE AND BREAK APART.

THE CONTINUAL MOVEMENT OF THE SEA

CAUSES THE REMAINS OF THE SHIP TO DISPERSE.

THE CONTINUAL DEPOSITS OF SAND AND SILT

WILL BURY THE TREASURE, LITERALLY ADDING

AN ADDITIONAL LAYER OF DIFFICULTY TO FINDING IT.

TREASURE HUNTERS RELY ON TECHNOLOGICAL TOOLS:

METAL DETECTORS, SUCTION DEVICES FOR REMOVING

LAYERS OF SAND, SONAR DEVICES AND GPS UNITS.

OF THE VARIOUS DEVICES IT IS THE GPS THAT IS BY FAR

THE MOST CRUCIAL PART OF THE HUNT.

AND GPS RELIES ON A COORDINATE SYSTEM.

AS YOU HAVE SEEN, LONGITUDE AND LATITUDE

ARE A COORDINATE SYSTEM IDEAL FOR NAVIGATION.

IN THE AREA OF TREASURE HUNTING,

THE DECIMAL VERSIONS OF LONGITUDE AND LATITUDE

IN COMBINATION WITH A CARTESIAN SYSTEM ARE USED.

THE LOCATION OF THE WRECK OF THE ATOCHA IS SHOWN HERE.

24 DEGREES, 32.673 MINUTES NORTH,

82 DEGREES, 20.806 MINUTES WEST.

RECALL THAT NORTH REFERS TO NORTH OF THE EQUATOR

AND WEST REFERS TO WEST OF THE PRIME MERIDIAN

IN GREENWICH, ENGLAND.

TRY THIS.

IF YOU HAVE ACCESS TO A COMPUTER WITH AN INTERNET

CONNECTION, GO TO GOOGLE MAPS OR ANOTHER WEBSITE

THAT WILL ACCEPT LONGITUDE AND LATITUDE COORDINATES.

INPUT THIS LONGITUDE AND LATITUDE AT THE ENTRY LINE

AND PRESS ENTER.

SELECT THE SATELLITE VIEW.

YOU CAN SEE WHERE THE SHIP'S DEBRIS WAS FIRST FOUND.

MAKE A NOTE OF THE MAP SCALE.

EVEN A SMALL REGION AROUND THE SITE

IS IN THE NEIGHBORHOOD OF 20 MILES

AND A 20 MILE BY 20 MILE AREA IS 400 SQUARE MILES.

YOU CAN BEGIN TO SEE WHY IT TOOK SO LONG

TO FIND THE WRECKAGE.

LET'S USE THE TI-NSPIRE TO CREATE A WAY

TO CONVERT FROM LONGITUDE AND LATITUDE

COORDINATES TO x,y COORDINATES.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A SPREADSHEET WINDOW.

MOVE THE CURSOR TO THE VERY TOP OF COLUMN A.

INPUT THE LABEL "DEGREES" AND PRESS TAB

TO GO TO THE TOP OF THE NEXT COLUMN.

INPUT THE LABEL "MINUTES/SECONDS."

PRESS ENTER.

MOVE THE CURSOR TO CELL A1.

START WITH WHAT WILL BE THE x COORDINATE.

LONGITUDE IS THE HORIZONTAL MEASUREMENT SO USE THAT.

SINCE THE LOCATION OF THE ATOCHA IS WEST OF

THE PRIME MERIDIAN, USE A NEGATIVE VALUE.

INPUT -82 IN CELL A1

AND PRESS THE TAB KEY TO GO TO CELL B1.

IN THIS COLUMN INPUT THE MINUTES AND SECONDS

OF LONGITUDE.

INPUT -20.806 AND PRESS ENTER.

MOVE THE CURSOR TO CELL A2.

HERE YOU WILL INPUT THE DEGREES OF LATITUDE.

SINCE THE LOCATION OF THE ATOCHA

IS NORTH OF THE EQUATOR USE A POSITIVE VALUE.

INPUT 24 AND PRESS THE TAB KEY TO GO TO CELL B2.

INPUT 32.673 AND PRESS ENTER.

NOW GO TO THE COLUMN HEADING

AT THE VERY TOP OF COLUMN C.

INPUT THE LABEL HEADING "DECIMAL" AND PRESS ENTER.

THE CURSOR IS NOW AT THE FORMULA LINE.

ANY FORMULAS YOU INPUT WILL AFFECT

ALL THE ENTRIES IN COLUMN C.

LET'S INPUT A FORMULA THAT TAKES THE DATA

FROM COLUMNS A AND B

AND CONVERTS IT INTO A DECIMAL COORDINATE.

SO AT THE FORMULA LINE, INPUT THIS EXPRESSION:

A+B DIVIDED BY 60.

PRESS ENTER.

YOU WILL NOW SEE THE CORRESPONDING LONGITUDE

AND LATITUDE COORDINATES WRITTEN AS DECIMALS.

LET'S GRAPH THIS COORDINATE AND HAVE IT

REPRESENT WHERE THE ATOCHA ORIGINALLY SANK.

WE WILL THEN GRAPH SOME ADDITIONAL POINTS

THAT REPRESENT LOCATIONS WHERE

SOME OF THE SHIP'S TREASURE WAS FOUND.

PRESS THE HOME KEY TO CREATE A NEW WORKSHEET.

SELECT A GRAPH WINDOW.

BY DEFAULT THERE IS A FUNCTION ENTRY LINE

WHICH, FOR THIS ACTIVITY, YOU WON'T NEED.

SO PRESS CONTROL AND G TO HIDE THE ENTRY LINE.

NEXT CHANGE THE WINDOW SETTINGS TO ONLY

FOCUS ON THE AREA NEAR WHERE THE ATOCHA SANK.

PRESS MENU AND UNDER WINDOW/ZOOM

CHANGE THE SETTINGS TO THIS:

PRESS THE TAB KEY TO MOVE FROM ONE DATA FLD

TO ANOTHER.

WHEN YOU HAVE CHANGED THE WINDOW SETTINGS

TAB YOUR WAY TO THE "OK" BUTTON.

PRESS ENTER.

NOW PLACE A POINT IN THE CENTER OF THE SCREEN.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

AND PRESS ENTER.

NOW DISPLAY THE COORDINATES.

MOVE THE POINTER ABOVE THE POINT

AND PRESS CONTROL AND MENU.

SELECT THE COORDINATES AND EQUATIONS OPTION.

YOU SHOULD NOW SEE THE COORDINATES OF THE POINT.

IF YOU NEED TO MOVE THE COORDINATES

MOVE THE POINTER ABOVE THE COORDINATES

AND PRESS AND HOLD THE CLICK KEY

TO SELECT THE COORDINATES.

USE THE NAVIGATION ARROWS TO MOVE THE COORDINATES

TO A NEW LOCATION AND THEN PRESS ENTER.

NOW CHANGE THE COORDINATES

TO THE ONES IN THE SPREADSHEET.

MOVE THE POINTER ABOVE THE TEXT OF

THE X COORDINATE AND PRESS ENTER.

THE TEXT FLD IS NOW EDITABLE.

PRESS THE CLEAR KEY TO REMOVE THE CURRENT

X COORDINATE AND REPLACE IT WITH THE DECIMAL

LONGITUDE VALUE FROM THE SPREADSHEET.

YOU CAN TOGGLE BACK TO THE SPREADSHEET

BY PRESSING CONTROL AND THE LEFT ARROW,

COPYING THE CONTENTS OF CELL C1

AND PASTING IT INTO THE COORDINATE FLD.

THE CONTROL LEFT OR RIGHT ARROW

TAKES YOU IN AND OUT OF THE DIFFERENT WINDOWS.

REPEAT WITH THE y COORDINATE.

MOVE THE POINTER ABOVE THE y COORDINATE

AND PRESS ENTER.

THEN INPUT OR PASTE IN THE DECIMAL LATITUDE VALUE.

YOUR SCREEN SHOULD LOOK LIKE THIS.

SINCE WE WILL BE GRAPHING OTHER COORDINATES

MAKE THIS POINT DISTINCTIVE

BY CHANGING ITS ATTRIBUTES.

MOVE THE POINTER ABOVE THE POINT

AND PRESS CONTROL AND MENU.

SELECT THE ATTRIBUTES OPTION.

USE THE LEFT OR RIGHT ARROWS

TO CHANGE THE APPEARANCE OF THE POINT.

THE X ICON MAY WORK BEST.

PRESS ENTER AFTER YOU HAVE SELECTED THE NEW ICON.

THE POINT ON SCREEN REPRESENTS WHERE

THE ATOCHA SANK BUT THE TREASURE WAS LOCATED

IN DOZENS OF LOCATIONS AROUND THIS POINT.

THIS TABLE SHOWS A HANDFUL OF POINTS

WHERE TREASURE WAS FOUND.

THE DATA POINTS ARE EXPRESSED IN LONGITUDE

AND LATITUDE COORDINATES.

USE THE SPREADSHEET TO CONVERT THESE

TO DECIMAL COORDINATES.

THEN GRAPH THESE POINTS.

FIRST PLACE POINTS ON THE GRAPH WINDOW.

DISPLAY THE COORDINATES AND THEN CHANGE THE VALUES

OF THE COORDINATES TO THE ONES FROM THE SPREADSHEET.

PAUSE THE VIDEO TO INPUT, CONVERT

AND GRAPH THE DATA POINTS.

WHEN YOU'RE DONE, TRY TO GET YOUR SCREEN

TO LOOK LIKE THIS.

YOU'LL SEE THAT THE TREASURE IS DISTRIBUTED

OVER A REGION AROUND THE POINT WHERE THE SHIP SANK.

AS EXPLAINED EARLIER THE DISPERSION OF THE WRECKAGE

HAS TO DO WITH THE MOVEMENT OF WATER

AND THE DECOMPOSITION OF THE WOOD ON THE SHIP.

HOW WIDE A REGION OF DISPERSAL IS THIS?

WE CAN USE THE SEGMENT TOOL TO CREATE A LINE

SEGMENT BETWEEN TWO POINTS AND THEN USE THE

MEASUREMENT TOOL TO MEASURE THE DISTANCE.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER TO THE SITE OF THE

SINKING OF THE ATOCHA.

PRESS ENTER.

THEN MOVE THE POINTER TO ONE OF THE POINTS

WHERE WRECKAGE WAS FOUND.

PRESS ENTER AGAIN.

NOW MEASURE THIS SEGMENT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE SEGMENT YOU JUST CREATED.

PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

MOVE THE POINTER TO THE SIDE OF THE SEGMENT

AND PRESS ENTER AGAIN

TO PLACE THE MEASUREMENT ON SCREEN.

YOUR MEASUREMENT SHOULD BE AROUND 1/10 OF A DEGREE.

ASSUME THAT THIS REPRESENTS ABOUT

HALF OF THE FULL DISPERSAL OF THE WRECKAGE.

SO THE RADIUS OF A CIRCLE THAT WOULD ENCOMPASS THE

FULL AREA WOULD HAVE A RADIUS OF 2/10 OF A DEGREE.

HOW LONG A DISTANCE IS THIS?

NEAR THE EQUATOR WHICH APPLIES TO THE FLORIDA KEYS

ONE DEGREE OF LATITUDE AND LONGITUDE

CORRESPONDS TO 60 MILES.

SO 2/10 OF A DEGREE CORRESPONDS TO 12 MILES.

A CIRCLE OF RADIUS 12 MILES HAS AN AREA

OF ROUGHLY 452 SQUARE MILES.

SO YOU CAN SEE THAT EVEN THOUGH THE LOCATION

OF A WRECKAGE MAY BE KNOWN,

THE FORCES OF DISPERSAL CAN MAKE FINDING

THE WRECKAGE MORE AND MORE DIFFICULT

AS MORE AND MORE TIME GOES ON.