Geometry Applications: Cylinders and Cones

Geometry Applications: Cylinders and Cones

[Music]

[Music]

THE CITY OF SHANGHAI IS CHINA'S MAJOR HUB

FOR FINANCE AND BUSINESS,

MUCH THE WAY MANHATTAN IS FOR THE U.S.

AND, MUCH LIKE NEW YORK CITY,

SHANGHAI HAS THE LOOK AND FEEL OF HIGH FINANCE.

THE FINANCIAL DISTRICT ALREADY HAS SOME

IMPRESSIVE TOWERS, YET THERE IS A NEW TOWER.

THE SHANGHAI TOWER, UNDER CONSTRUCTION,

WHICH, WHEN COMPLETED IN 2014,

WILL BE THE TALLEST BUILDING IN CHINA

AND THE SECOND TALLEST BUILDING IN THE WORLD.

THIS STREAMLINED TOWER HAS AN IRREGULAR SHAPE,

BUT UNDERNEATH THE FLOWING EXTERIOR

IS A SOLID GEOMETRIC BASE.

THERE IS A STACK OF EIGHT CYLINDRICAL SHAPES.

HERE IS A CUT-AWAY VIEW TO REVEAL THE EIGHT CYLINDERS.

NOTICE THAT EACH SUCCEEDING CYLINDER

IS NARROWER THAN THE ONE RIGHT BELOW IT.

THIS IS BECAUSE THE SIDE OF THE TOWER

IS ALONG A LINE THAT IS TWO DEGREES

FROM THE VERTICAL AXIS OF THE TOWER.

PUT A LITTLE MORE SUBTLY, EACH CYLINDER INCREASES

IN HEIGHT FROM THE ONE RIGHT BELOW IT

AND THE COMBINATION OF BOTH CHANGES IN DIMENSION

MAKES THE TOWER LOOK TALL AND SLEEK.

PART OF THE REASON THAT THE TOWER IS THINNER AT THE TOP

HAS TO DO WITH GEOGRAPHY.

THIS REGION OF CHINA EXPERIENCES TYPHOONS

WHICH HAVE VERY HIGH WINDS.

A BUILDING THAT IS THINNER AT THE TOP

IS LESS SUSCEPTIBLE TO THE FORCE OF THESE WINDS.

IN FACT, THE OUTER LAYER OF THE TOWER

SWIRLS AROUND THE CORE OF THE TOWER IN SUCH A WAY

AS TO DECREASE THE FORCE OF THE WIND ON THE TOWER.

NOT ONLY THAT, THERE ARE WIND TURBINES LOCATED

WITHIN THE TOWER TO USE THE POWER OF WIND

TO GENERATE ELECTRICITY.

THE SHANGHAI TOWER IS AN EXAMPLE OF

GREEN ARCHITECTURE.

THE SWIRLING OUTER LAYER IS ALSO A COLLECTOR

OF RAINWATER THAT IS RECYCLED FOR USE

IN THE AIR CONDITIONING AND HEATING SYSTEMS.

BUT THE HEART OF THE TOWER IS THE STACK OF CYLINDERS

WHICH IS WHERE OFFICE AND LIVING SPACES

WILL BE HOUSED.

AND GOING FROM ONE TIER TO THE NEXT,

THE WIDTH OF THE CYLINDER DECREASES BY 8.2%

AND ITS HEIGHT INCREASES BY 4.1%.

WE CAN USE THESE VALUES TO DETERMINE THE VOLUME

AND THEREFORE THE WEIGHT THAT EACH SECTION CHANGES

FROM ONE LEVEL TO THE NEXT.

OBVIOUSLY THE HEAVIER LEVELS ARE AT THE BOTTOM

BUT HOW MUCH LIGHTER DO THE SUCCEEDING TIERS GET?

A CYLINDER IS A THREE DIMENSIONAL FIGURE

WITH A CIRCULAR BASE AND A RECTANGULAR SIDE.

A NET FOR A CYLINDER CLEARLY SHOWS THE TWO CIRCLES THAT

DEFINE THE BASE AND THE TOP AND THE RECTANGULAR SIDE.

THE SURFACE AREA OF THE CYLINDER IS MADE UP OF

INDIVIDUAL AREAS OF THE CIRCLES AND THE RECTANGLE.

SUPPOSE THE CIRCLE HAS RADIUS r.

SINCE THE TOP SIDE OF THE RECTANGLE WRAPS AROUND

THE CIRCLE, THEN THE RECTANGLE'S SIDE LENGTH

IS THE SAME AS THE CIRCLE'S PERIMETER.

THE OTHER SIDE LENGTH OF THE RECTANGLE

CORRESPONDS TO THE HEIGHT OF THE CYLINDER.

SO THE SURFACE AREA OF THE CYLINDER IS 2 Pi r SQUARED

PLUS 2 Pi rh.

THE VOLUME OF THE CYLINDER IS Pi r SQUARED h.

LET'S USE THE VARIABLES r AND h FOR THE BOTTOM

CYLINDRICAL PORTION OF THE SHANGHAI TOWER.

EACH SUCCEEDING CYLINDER DECREASES ITS WIDTH BY 8.2%

AND INCREASES ITS HEIGHT BY 4.1%.

SINCE THE DIAMETER OF THE CIRCULAR BASE

DECREASES BY 8.2%, THIS MEANS THAT THE RADIUS

DECREASES BY HALF THAT AMOUNT, OR 4.1%.

SO LEVEL TWO IS Pi r SQUARED h TIMES 1.041 TIMES 0.959.

FOR ANY TIER i, THE VOLUME OF THAT CYLINDER

IS FOUND WITH THIS EXPRESSION.

THE TOTAL VOLUME OF ALL THE CYLINDRICAL SECTIONS

IS FOUND USING THIS EXPRESSION.

LET'S USE THE TI-NSPIRE TO CALCULATE THIS VALUE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A CALCULATOR WINDOW.

UNLESS YOU ARE USING A TI-NSPIRE CAS,

THEN YOU CANNOT USE THE VARIABLES r AND h.

BUT SINCE THESE VARIABLES ARE FOUND IN EVERY TERM,

THEY CAN BE LEFT OUT.

IN FACT, THE EXPRESSION YOU WILL BE INPUTTING IS THIS:

PRESS THE LIBRARY BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS THE NUMBER 4 TO BRING UP THE TAB

THAT HAS THE SUMMATION SYMBOL.

MOVE THE POINTER UNTIL THE SIGMA SYMBOL IS HIGHLIGHTED.

PRESS ENTER.

INPUT THE LETTER i IN THE FIRST FLD

AND PRESS THE RIGHT ARROW KEY.

INPUT THE NUMBER 1.

PRESS THE RIGHT ARROW.

INPUT THE NUMERICAL EXPRESSION AS SHOWN HERE.

THEN PRESS THE UP ARROW AND INPUT THE NUMBER 8.

PRESS ENTER TO CALCULATE THE RESULT.

YOU'LL SEE THAT THE SUM IS ABOUT 6.9.

CONTRAST THIS TO A SITUATION IN WHICH EIGHT CYLINDERS

THE SAME SIZE AS THE BASE CYLINDER ARE STACKED.

THE SUM OF THE VOLUMES OF THE STACKED CYLINDERS

WOULD BE EIGHT TIMES THE VALUE OF THE BASE CYLINDER.

IN OTHER WORDS, THE SHANGHAI TOWER HAS 14% LESS VOLUME

AND CORRESPONDINGLY LESS WEIGHT

WHICH HELPS IN KEEPING THE COST OF MATERIALS DOWN.

HOW IS THE WEIGHT OF THE TOWER DISTRIBUTED?

LET'S ANALYZE THIS IN A SPREADSHEET.

PRESS THE HOME KEY AND SELECT

A LIST AND SPREADSHEET WINDOW.

PRESS ENTER.

MOVE THE CURSOR TO THE VERY TOP OF COLUMN A

AND THE COLUMN HEADING "TIER".

PRESS THE DOWN ARROW.

YOUR CURSOR SHOULD BE ABOVE CELL A1

AND RIGHT BELOW THE COLUMN HEADING.

THIS IS THE FORMULA LINE FOR INSERTING SPREADSHEET

FORMULAS THAT APPLY TO THE ENTIRE COLUMN.

CREATE A CONSECUTIVE SEQUENCE OF NUMBERS

FROM ONE TO EIGHT.

PRESS MENU AND UNDER DATA SELECT GENERATE SEQUENCE.

AT THE DIALOGUE BOX INPUT N+1 AND PRESS TAB.

AT THE NEXT FLD INPUT 1 AS THE START VALUE

FOR THE SEQUENCE.

IN OTHER WORDS, THE SEQUENCE STARTS AT N=0.

PRESS TAB AND FOR THE MAXIMUM NUMBER OF TERMS

INPUT 7, NOT 8, KEEPING IN MIND THAT THE SEQUENCE

STARTS AT N=0 AND WILL INCLUDE EIGHT TERMS.

PRESS ENTER.

YOU'LL SEE A CONSECUTIVE LIST OF NUMBERS

FROM ONE TO EIGHT.

NEXT, MOVE TO THE FORMULA LINE FOR COLUMN B,

THE CELL BELOW THE COLUMN HEADING AND ABOVE CELL B1.

INPUT THIS EXPRESSION AT THE FORMULA LINE.

PRESS ENTER.

FOR EACH CELL IN COLUMN B, THIS FORMULA

USES THE CORRESPONDING VALUE IN COLUMN A

TO CALCULATE THE VOLUME.

NOTICE THAT THE VALUES OF THE VOLUME

GRADUALLY DECREASE IN GOING FROM

CYLINDRICAL REGION ONE TO EIGHT.

HOW DOES THE CHANGE IN DIMENSIONS

AFFECT THE SURFACE AREA OF THE CYLINDERS?

RECALL THAT THE FORMULA FOR FINDING SURFACE AREA

OF A CYLINDER IS 2 Pi r SQUARED PLUS 2 Pi rh.

FOR NOW LET'S FOCUS ON THE SECOND TERM,

WHICH IS THE SURFACE AREA OF THE SIDE OF THE CYLINDER.

THE AREA OF THE SIDE OF THE CYLINDER

FOR TIER ONE IS 2 Pi rh.

FOR LEVEL 2 THE AREA IS 2 Pi rh TIMES 0.959 TIMES 1.041.

FOR ANY LEVEL i

THE AREA OF THE SIDE OF THE CYLINDER IS THIS.

CREATE A FORMULA IN THE CALCULATOR WINDOW

TO FIND THE SIDE SURFACE AREA.

PRESS THE LIBRARY BUTTON AND SELECT THE SIGMA TEMPLATE.

INPUT THE LETTER i IN THE FIRST FLD

AND PRESS THE RIGHT ARROW KEY.

INPUT THE NUMBER 1.

PRESS THE RIGHT ARROW.

INPUT THE NUMERICAL EXPRESSION AS SHOWN HERE.

THEN PRESS THE UP ARROW AND INPUT THE NUMBER 8.

PRESS ENTER TO CALCULATE THE RESULT.

YOU'LL SEE THAT THE SURFACE AREA IS ALMOST 8.

SO WHILE THE TOWER HAS 14% LESS VOLUME AND WEIGHT,

IT HAS ALMOST THE SAME FLOOR SPACE

AS IF THE BUILDING WERE NOT TAPERED.

THIS IS AN EFFICIENT WAY TO CREATE A LIGHTER WEIGHT

BUILDING WITHOUT SACRIFICING SPACE.

BY CAREFULLY MANAGING THE CHANGE IN DIMENSIONS

OF A BUILDING FROM ONE LEVEL TO ANOTHER,

AN ARCHITECT COULD MAKE SUBSTANTIAL CHANGES

TO THE SURFACE AREA OR VOLUME OF A BUILDING.

THIS IS CLEARLY SEEN IN COMPARING THE GRAPHS

OF THE VOLUME AND SURFACE AREA EQUATIONS.

PRESS THE HOME KEY AND SELECT THE GRAPH WINDOW.

PRESS ENTER.

AT FUNCTION ENTRY LINE F1 INPUT THIS FUNCTION

WHICH REPRESENTS THE CHANGE IN VOLUME.

PRESS THE DOWN ARROW.

AT THE F2 FUNCTION ENTRY LINE INPUT THIS FUNCTION

WHICH REPRESENTS THE CHANGE IN THE SIDE SURFACE AREA.

PRESS ENTER.

TO GET A BETTER VIEW OF THE GRAPH, PRESS MENU

AND UNDER WINDOW/ZOOM SELECT ZOOM-FIT.

YOUR GRAPH SHOULD LOOK LIKE THIS.

BOTH GRAPHS ARE DECREASING FUNCTIONS

WHICH MEANS THAT FOR HIGHER VALUES OF X

THE GRAPH IS AT A LOWER Y COORDINATE.

BUT NOTICE THAT THE GRAPH OF THE VOLUME

DECREASES AT A MUCH FASTER RATE

THAN THE SIDE SURFACE AREA GRAPH.

THIS MEANS THE SHANGHAI SHEDS VOLUME AND THEREFORE

WEIGHT AT A FASTER RATE THAN IT SHEDS SURFACE AREA.

THIS GRAPH CONFIRMS THAT THE ARCHITECTS

ARE ABLE TO GET MORE OFFICE AND LIVING SPACE

WITHOUT INCREASING THE WEIGHT OF THE BUILDING.

THE SLOWER DECREASE IN THE SURFACE AREA

ALSO HELPS WITH THE OVERALL HEIGHT OF THE TOWER.

AT THE TIME THIS BUILDING WAS BEING PLANNED,

ITS HEIGHT WOULD HAVE MADE IT THE TALLEST BUILDING

IN THE WORLD, AND PART OF THIS WOULD HAVE BEEN DUE TO

THE INCREASING HEIGHT OF EACH CYLINDRICAL SECTION

WHICH WOULD HAVE ADDED JUST ENOUGH HEIGHT

TO HAVE PUSHED IT OVER THE TOP.

HOWEVER, SINCE THE ORIGINAL PLANNING OF THIS BUILDING

THE BURJ KHALIFA TOWER IN DUBAI

STANDS AS THE TALLEST BUILDING.

NOTICE THAT THIS TOWER, TOO,

HAS SOME OF THE FEATURES OF THE SHANGHAI TOWER:

THE CYLINDRICAL SECTIONS IN GOING FROM TOP TO BOTTOM

INCREASE IN HEIGHT AND DECREASE IN WIDTH.

THIS IS AN EFFECTIVE COMBINATION

FOR CREATING A SUPER TALL SKYSCRAPER.

BUILDINGS LIKE THE SHANGHAI TOWER AND THE BURJ KHALIFA

ARE SO MASSIVE THAT THEY ARE MINIATURE CITIES

AND SO THE ARCHITECTURE OF BUILDINGS LIKE THESE

RELY ON GREEN SUSTAINABLE TECHNIQUES

TO MAKE THEM MORE LIVABLE AND MANAGEABLE.

ONE OF THE LIKELY CHARACTERISTICS

OF FUTURE SUPER TALL SKYSCRAPERS

IS THE TAPERED LOOK - WIDER AT THE BOTTOM

AND COMING TO A POINT NEAR THE TOP.

THE USE OF CYLINDRICAL FORMS IN COMBINATION

WITH A TAPERING DESIGN TENDS TO

GIVE THESE BUILDINGS THE LOOK OF A CONE.

COMPARING THE VOLUME AND SURFACE AREA

OF A CONE AND CYLINDER SHOWS WHY.

THE CONE HAS A THIRD OF THE VOLUME OF THE CYLINDER

AND SIGNIFICANTLY LESS SURFACE AREA.

AND OF COURSE AS A BUILDING

APPROXIMATES THE SHAPE OF A CONE

IT ALSO BEGINS TO EVOKE THE SHAPE OF A PYRAMID.

SHANGHAI IS A CITY IN TRANSITION

AND CHINA IS A COUNTRY IN TRANSITION.

AS SKYSCRAPERS REACH FOR THE HEAVENS,

ARCHITECTS TAP INTO TECHNIQUES

THAT GO BACK TO THE AWE-INSPIRING WORKS

OF THE ANCIENT WORLD.