Geometry Applications: Volume and Density

[Music]

ON APRIL 10, 1912 THE TITANIC WAS LAUNCHED.

AT THE TIME IT WAS THE WORLD'S LARGEST

PASSENGER STEAMSHIP.

IT WAS SCHEDULED TO SAIL FROM SOUTHAMPTON, ENGLAND

TO NEW YORK CITY, MAKING A FEW STOPS ALONG THE WAY.

BUT FOUR DAYS INTO THE VOYAGE

IN THE MIDDLE OF THE ATLANTIC OCEAN

THE TITANIC HIT AN ICEBERG

THAT PUNCTURED THE HULL OF THE SHIP

CAUSING IT TO TAKE ON WATER.

IN LESS THAN THREE HOURS

THE SHIP WAS SO WATERLOGGED THAT IT SANK.

NEARLY 2,000 PEOPLE DIED.

AS WE APPROACH THE 100TH ANNIVERSARY OF THIS

TRAGIC SHIPWRECK, ONE THING HASN'T CHANGED:

WHY THE TITANIC SANK.

BUT BEFORE ADDRESSING THAT, IT'S IMPORTANT

TO UNDERSTAND WHY A SHIP OF THE TITANIC'S SIZE

COULD EVEN FLOAT IN THE FIRST PLACE.

AFTER ALL, THE SHIP WAS MADE UP OF 46,000 TONS

OF METAL, WOOD AND OTHER MATERIALS.

THIS ILLUSTRATION SHOWS THE COMPARATIVE SIZE

OF THE TITANIC.

CLEARLY THIS WAS A MASSIVE SHIP.

BUT HOWEVER MUCH MASS A SHIP MIGHT HAVE,

SO LONG AS ITS DENSITY IS LESS THAN

THE DENSITY OF WATER, IT WILL FLOAT.

DENSITY IS THE RATIO OF THE MASS OF AN OBJECT

TO THE VOLUME OF THE OBJECT,

WHICH IS THE SCIENTIFIC DEFINITION OF DENSITY.

LET'S LOOK AT THE GEOMETRY BEHIND DENSITY.

HERE IS A CUBE.

IT HAS MASS M AND VOLUME V.

AS THE MASS INCREASES SO DOES THE VOLUME.

THE INCREASE IN THE MASS AND VOLUME

HAPPENS AT A CONSTANT RATE

AND CAN BE SUMMARIZED WITH THIS EQUATION: M=cV.

IN THE EQUATION, c IS A CONSTANT.

AND THIS EQUATION IS AN EXAMPLE OF A

DIRECT VARIATION.

THE GRAPH OF A DIRECT VARIATION

IS A LINEAR FUNCTION THAT CROSSES THE ORIGIN.

THE CONSTANT c IS CALLED THE CONSTANT OF VARIATION.

WHEN YOU TAKE THE RATIO OF M OVER V

THEN THE CONSTANT c IS ALSO THE SLOPE

OF THE LINEAR FUNCTION.

THIS CONSTANT OF VARIATION, THE SLOPE,

IS WHAT SCIENTISTS REFER TO AS THE DENSITY.

FOR SIMPLICITY, IN THE METRIC SYSTEM

THE DENSITY OF WATER IS EQUAL TO 1.

ANY MATERIAL THAT HAS A DENSITY GREATER THAN 1,

OR A LINEAR GRAPH WITH A SLOPE GREATER THAN 1

IS DENSER THAN WATER AND WILL SINK.

ANY MATERIAL THAT HAS A DENSITY LESS THAN 1

OR A LINEAR GRAPH WITH A SLOPE LESS THAN 1

IS LESS DENSE THAN WATER AND WILL FLOAT.

THE EQUATION FOR DENSITY IS A FUNCTION.

THE VARIABLE IS V AND FOR ANY GIVEN SUBSTANCE

M IS A CONSTANT.

THE FUNCTION IS WRITTEN THIS WAY AND IS READ THIS WAY:

D OF V EQUALS M OVER V.

RETURNING TO THE EXAMPLE OF THE CUBE SHAPED OBJECT

OF MASS M, WE KNOW THAT THE VOLUME OF A CUBE

IS S CUBED WHERE S IS THE LENGTH OF

ANY OF THE EDGES OF THE CUBE.

SO FOR A CUBE WE GET THE FUNCTION

D EQUALS M OVER S CUBED.

THIS IS AN EXAMPLE OF A RATIONAL FUNCTION

WHOSE GRAPH HAS THIS SHAPE.

LET'S EXPLORE THIS GRAPH ON THE NSPIRE AND SEE

WHAT WE CAN LEARN ABOUT DENSITY AND THE TITANIC.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

YOU'LL BE GRAPHING y=m OVER x SQUARED

WHERE M CAN TAKE ON DIFFERENT VALUES.

TO DO THAT CREATE A SLIDER.

PRESS MENU AND UNDER ACTIONS SELECT INSERT SLIDER.

YOU'LL SEE THE SLIDER APPEAR ON THE UPPER LEFT CORNER OF

THE SCREEN WITH THE VARIABLE NAME ALREADY HIGHLIGHTED.

CHANGE THE VARIABLE TO m AND PRESS ENTER.

NEXT PRESS TAB TWICE TO GO TO THE FUNCTION ENTRY LINE.

PRESS CONTROL AND THE DIVISION SYMBOL TO CREATE

A PLACEHOLDER FOR THE RATIONAL EXPRESSION.

INPUT THE LETTER m IN THE NUMERATOR.

PRESS THE DOWN ARROW TO GO TO THE DENOMINATOR

AND INPUT X CUBED, THEN PRESS ENTER.

BY DEFAULT THE VALUE OF m IS 5

AND IT RANGES IN VALUE FROM 0 TO 10.

MOVE THE POINTER ABOVE THE SLIDER.

PRESS AND HOLD THE CLICK KEY

UNTIL THE OPEN HAND BECOMES A CLOSED HAND.

USE THE LEFT AND RIGHT ARROW KEYS

TO CHANGE THE VALUE OF m.

YOU'LL SEE THAT AS THE VALUE OF m INCREASES

THE GRAPH SHIFTS TO THE RIGHT.

PRESS CONTROL AND G TO BRING UP THE

FUNCTION ENTRY LINE FOR f2.

INPUT 1 AND PRESS ENTER.

THIS VALUE REPRESENTS THE DENSITY OF WATER.

ANY VALUES OF y ABOVE THE HORIZONTAL LINE

REPRESENT A CONFIGURATION TOO DENSE TO FLOAT.

NOW COMPARE THE GRAPH

AS THE VALUE OF THE VOLUME INCREASES.

CREATE A NEW SLIDER.

PRESS MENU AND UNDER ACTIONS SELECT INSERT SLIDER.

THIS SLIDER WILL APPEAR ABOVE THE FIRST SLIDER

WITH THE TEXT FLD HIGHLIGHTED.

INPUT THE LETTER C AND PRESS ENTER.

PRESS ESCAPE AND MOVE THE POINTER

ABOVE THE SECOND SLIDER.

PRESS AND HOLD THE CLICK KEY

TO SELECT THE SECOND SLIDER.

MOVE IT TO THE OTHER SIDE OF THE SCREEN

AND PRESS ESCAPE.

PRESS CONTROL AND G

TO BRING UP THE FUNCTION ENTRY LINE FOR f3.

PRESS CONTROL AND THE DIVISION SYMBOL TO CREATE

A PLACEHOLDER FOR ANOTHER RATIONAL EXPRESSION.

INPUT M IN THE NUMERATOR AND PRESS THE DOWN ARROW.

INPUT THE EXPRESSION C TIMES X CUBED.

THIS REPRESENTS A SITUATION WHERE THE MASS OF THE OBJECT

IS THE SAME BUT THE VOLUME CHANGES.

PRESS ENTER.

FOR VALUES OF C GREATER THAN 1

THE DENSITY DECREASES COMPARED TO THE OTHER GRAPH.

SO SUPPOSE YOU HAVE A CONTAINER

MADE OF A HEAVY MATERIAL.

IN ORDER FOR IT TO FLOAT

THE VOLUME OF AIR INSIDE THE CONTAINER

MUST BE LARGE ENOUGH THAT THE RATIO OF

MASS TO VOLUME IS LESS THAN OR EQUAL TO 1.

THIS IS WHY A SHIP MADE OUT OF METAL

CAN STILL FLOAT ON WATER.

THE HULL OF THE SHIP IS BASICALLY A HOLLOWED OUT

VOLUME OF AIR THAT KEEPS THE DENSITY OF THE SHIP

LESS THAN OR EQUAL TO THE DENSITY OF WATER.

RETURNING TO THE TITANIC, THE HULL OF THAT SHIP

CAN BE APPROXIMATED BY A TRIANGULAR PRISM.

A TRIANGULAR PRISM HAS TWO TRIANGULAR FACES

AND THREE RECTANGULAR SIDES.

A NET FOR A TRIANGULAR PRISM LOOKS LIKE THIS.

THE VOLUME OF A RECTANGULAR PRISM

IS THE AREA OF THE TRIANGULAR BASE

TIMES THE LENGTH OF THE RECTANGULAR LENGTH.

THE FORMULA V EQUALS A TIMES L

CAN BE USED TO CALCULATE THE VOLUME.

BUT WE ALSO KNOW THE AREA OF A TRIANGLE

IS ONE HALF THE BASE TIMES THE HEIGHT.

SO THE VOLUME FORMULA CHANGES TO

V EQUALS ONE HALF B TIMES H TIMES L.

CONTRAST THIS TO A RECTANGULAR PRISM

WHICH HAS TWO RECTANGULAR FACES.

IF THE TWO PRISMS, ONE RECTANGULAR

AND ONE TRIANGULAR HAVE SIMILAR DIMENSIONS,

THE LENGTH AND THE HEIGHT,

THE RECTANGULAR PRISM WILL HAVE MORE VOLUME.

IF A SHIP LIKE THE TITANIC NEEDS TO MAXIMIZE ITS VOLUME

WHY DOESN'T IT HAVE A HULL IN THE SHAPE OF A

RECTANGULAR PRISM INSTEAD OF A TRIANGULAR PRISM?

THE ANSWER HAS TO DO WITH THE PHYSICS OF SAILING.

A TRIANGULAR PRISM SHAPED HULL WILL SLICE THROUGH THE

WATER AND MOVE MUCH FASTER THAN A RECTANGULAR HULL.

THE TITANIC WAS CALLED THIS FOR A REASON.

ITS DIMENSIONS WERE:

WHENEVER YOU PLACE AN OBJECT IN WATER

IT DISPLACES AN AMOUNT OF WATER

EQUAL TO THE WEIGHT OF THE OBJECT.

THE TITANIC'S DISPLACEMENT WAS 46,000 TONS.

THE HEIGHT LISTED HERE IS JUST FOR THE HEIGHT OF THE

TRIANGULAR PRISM SECTION THAT MAKES UP THE HULL.

LET'S USE THE TI-NSPIRE TO CALCULATE THE VOLUME

AND DENSITY OF THE TITANIC'S HULL.

CREATE A CALCULATOR WINDOW.

PRESS THE HOME KEY AND SELECT CALCULATOR.

TO CALCULATE THE VOLUME OF THE

TRIANGULAR PRISM THAT WE ARE USING

TO APPROXIMATE THE VOLUME OF THE HULL

INPUT THE EXPRESSION 0.5 X 270 X 28 X 30

AND RATHER THAN PRESSING ENTER

ASSIGN THIS VALUE TO A VARIABLE.

PRESS THE CONTROL BUTTON AND THE VAR KEY.

YOU WILL SEE AN ARROW POINTING TO THE RIGHT.

INPUT THE VARIABLE NAMED VOLUME AND PRESS ENTER.

YOU HAVE ASSIGNED THE CALCULATED VALUE

OF THE VOLUME TO VARIABLE.

THE VOLUME THAT YOU CALCULATED

IS IN METERS CUBED.

THE UNITS OF DENSITY

THAT DEFINE WATER WITH A DENSITY OF 1

IS GRAMS PER CENTIMETERS CUBED.

WE NEED TO CONVERT FROM METERS CUBED

TO CENTIMETERS CUBED.

SO MULTIPLY THE RESULT BY 100 CUBED.

INPUT THE EXPRESSION VOLUME TIMES 100 CUBED

AND ASSIGN IT TO THE SAME VARIABLE VOLUME.

PRESS ENTER.

ASSIGN THE VALUE 46,000 TO A NEW VARIABLE CALLED MASS.

NOW WE NEED TO CONVERT TONS TO GRAMS.

THE CONVERSION FACTOR IS 1.01605 TIMES 10 TO THE SIXTH.

INPUT THIS EXPRESSION INTO THE CALCULATOR WINDOW.

MASS TIMES 1.01605 TIMES 10 TO THE SIXTH

AND ASSIGN THIS VALUE BACK INTO THE MASS VARIABLE.

PRESS ENTER.

YOU ARE NOW READY TO CALCULATE THE DENSITY.

INPUT THE EXPRESSION MASS OVER VOLUME

AND ASSIGN IT TO A NEW VARIABLE: DENSITY.

PRESS ENTER.

YOU'LL SEE THAT THE DENSITY OF THE TITANIC

BEFORE IT HIT THE ICEBERG WAS ROUGHLY 0.4 WHICH,

BECAUSE THE VALUE IS LESS THAN 1,

WOULD HAVE MEANT THAT THE SHIP WAS FLOATING.

WHICH IT WAS.

BUT ONCE THE TITANIC HIT THE ICEBERG

WATER STARTED FLOODING THE HULL.

WHEN A SHIP STARTS TAKING ON WATER ITS DENSITY CHANGES.

BUT IT CHANGES QUICKLY BECAUSE BOTH THE MASS

AND THE VOLUME ARE CHANGING.

THE INCREASING AMOUNT OF WATER INCREASES THE

OVERALL MASS OF THE SHIP WHICH INCREASES THE DENSITY.

THE INCREASE IN THE AMOUNT OF WATER

COMES AT THE EXPENSE OF THE VOLUME OF AIR IN THE HULL.

THE VOLUME DECREASES

MAKING THE DENSITY INCREASE EVEN MORE.

THE COMBINATION OF BOTH VARIABLES CHANGING RAPIDLY

INCREASES THE DENSITY.

IN FACT FOR ANY VOLUME DECREASE X

THAT AMOUNT OF VOLUME IS CONVERTED INTO

A CORRESPONDING AMOUNT OF WATER.

FOR ANY CUBIC CENTIMETER OF WATER MULTIPLY BY .001

TO FIND THE CORRESPONDING MASS OF WATER.

FOR EXAMPLE SUPPOSE ONE-TENTH OF THE VOLUME

OF THE HULL IS FILLED WITH WATER.

THIS EXPRESSION IS THE EXPRESSION

FOR THE NEW DENSITY.

INPUT THIS EXPRESSION INTO THE CALCULATOR WINDOW

AND PRESS ENTER.

THE NEW DENSITY IS ABOUT 0.46.

SO THE DENSITY HAS GONE UP

BUT IT IS STILL MUCH LOWER THAN ONE.

SO THE SHIP WOULD NOT HAVE SUNK

TAKING IN THIS AMOUNT OF WATER.

AT WHAT PERCENTAGE OF THE VOLUME HAS THE SHIP

TAKEN ON TOO MUCH WATER AND WILL SINK?

WE CAN USE A GENERAL FORM OF THE EXPRESSION.

IN THIS EXPRESSION THE X OVER 100 TERM REPRESENTS THE

PERCENTAGE OF THE VOLUME CONVERTED TO WATER.

LET'S GRAPH THE FUNCTION BASED ON THIS EXPRESSION

AND SEE WHERE IT INTERSECTS THE GRAPH OF y=1.

PRESS THE HOME KEY AND SELECT A GRAPH WINDOW.

AT THE f1 ENTRY LINE, INPUT 1

AND PRESS THE DOWN ARROW.

AT THE f2 FUNCTION ENTRY LINE INPUT THIS EXPRESSION.

PRESS CONTROL AND THE DIVISION SYMBOL TO CREATE

A PLACEHOLDER FOR THE RATIONAL EXPRESSION.

AFTER YOU HAVE INPUT THE FUNCTION PRESS ENTER.

CHANGE THE WINDOW SETTINGS, PRESS MENU

AND UNDER WINDOW/ZOOM CHANGE THE WINDOW SETTINGS

TO XMIN EQUALS 0, XMAX EQUALS 100,

YMIN EQUALS 0 AND YMAX EQUALS 5.

PRESS OKAY.

NOW FIND WHERE THE TWO GRAPHS INTERSECT.

PRESS MENU AND UNDER POINTS AND LINES

SELECT INTERSECTION POINT.

MOVE THE POINTER ABOVE EACH GRAPH

AND PRESS ENTER EACH TIME.

YOU'LL SEE THE INTERSECTION POINT AS SLIGHTLY ABOVE 41%

SO THE TITANIC HAD TO TAKE ON A PERCENTAGE OF THE

HULL'S VOLUME GREATER THAN 41% BEFORE IT WOULD SINK.

NOW THE TITANIC DID HAVE SEPARATE

WATERTIGHT COMPARTMENTS SO THAT

IF THERE WERE A HOLE IN THE HULL OF THE SHIP

NOT ALL OF THE HULL WOULD BE FLOODED WITH WATER.

CONTAINING THE AMOUNT OF WATER WOULD HAVE

PREVENTED THE SHIP FROM SINKING.

SO WHY DID IT SINK?

AS IT TURNS OUT, OF THE 16 WATERTIGHT COMPARTMENTS

ON THE TITANIC SIX OF THEM WERE DAMAGED

AS THE ICEBERG SCRAPED THE SIDE OF THE HULL.

ENOUGH COMPARTMENTS WERE DAMAGED TO LET IN

ENOUGH WATER FOR THE SHIP TO REACH THE THRESHOLD

WHERE IT WOULD SINK.

HAD THE ICEBERG DAMAGED FEWER OF THE COMPARTMENTS

IT'S POSSIBLE THAT THE TITANIC WOULD HAVE

TAKEN ON WATER BUT NOT SUNK.

SO AS YOU CAN SEE SUBTLE CHANGES IN THE MEASURE

OF DENSITY CAN HAVE ENORMOUS CONSEQUENCES.