Geometry Applications: Surface Area to Volume Ratio

Geometry Applications: Surface Area to Volume Ratio

[Music]

[Music]

THE CITIGROUP BUILDING IN NEW YORK CITY

HAS A DISTINCTIVE OUTLINE.

FROM TOP TO BOTTOM THERE ARE SOME CLEVER INNOVATIONS.

AT THE TIME THE BUILDING WAS COMMISSIONED

THERE WAS A CHURCH WHERE THE TOWER WOULD STAND

AND AS PART OF THE ARRANGEMENT

THE TOWER NEEDED TO LEAVE ROOM FOR THE CHURCH.

THIS IS WHY THE BASE OF THE TOWER CONSISTS OF

FOUR COLUMNS THAT CLEAR A SPACE

ON THE GROUND FLOOR FOR THE CHURCH.

THE TOP OF THE TOWER HAS A SLANTED ROOF WHERE SOLAR

PANELS ARE INSTALLED AS AN ENERGY SAVING MEASURE.

THE SHAPE OF THE TOWER IS A COMBINATION

OF A SQUARE PRISM AND A TRIANGULAR PRISM.

A SQUARE PRISM IS A TYPE OF RECTANGULAR PRISM

WHERE THE TOP AND BOTTOM FACES ARE SQUARES.

THE TRIANGULAR PRISM IS THE ROOF SECTION OF THE TOWER.

WHILE NOT THE TALLEST SKYSCRAPER IN NEW YORK CITY,

THE CITIGROUP TOWER STILL HAS TO CONTEND WITH

A PROBLEM COMMON TO TALL BUILDINGS:

THE CONTINUAL LOSS OF HEAT.

KEEPING A BUILDING LIKE THIS COOL IN THE SUMMER

AND WARM IN THE WINTER CAN GET EXPENSIVE IF THE TOWER

CHANGES ITS INTERNAL TEMPERATURE TOO QUICKLY.

THE ABILITY FOR A BUILDING TO RETAIN HEAT

IS AFFECTED BY ITS SHAPE.

ONE OF THE BEST INDICATORS OF THE

POTENTIAL FOR HEAT LOSS IS TO FIND THE RATIO

OF THE BUILDING SURFACE AREA TO VOLUME.

TO GET A BETTER UNDERSTANDING

OF WHY THIS RATIO WAS IMPORTANT,

LET'S LOOK AT EXAMPLES FROM NATURE.

SNAKES ARE LONG AND THIN.

THIS SHAPE GIVES THEM A LOT OF SURFACE AREA

BUT NOT A LOT OF VOLUME.

SO THE RATIO OF SURFACE AREA TO VOLUME

IS RELATIVELY LARGE.

WITH MORE SURFACE AREA RELATIVE TO VOLUME

A SNAKE WILL LOSE HEAT QUICKLY.

THIS IS WHY SNAKES SPEND TIME IN THE SUN

ABSORBING HEAT TO MAKE UP FOR WHAT THEY LOST.

ON THE OTHER HAND, A POLAR BEAR HAS A

LARGER VOLUME COMPARED TO ITS SURFACE AREA.

THE RATIO OF SURFACE AREA TO VOLUME

FOR THE POLAR BEAR IS RELATIVELY SMALL WHICH

ALLOWS THE POLAR BEAR TO RETAIN HEAT LONGER.

THIS IS IMPORTANT SINCE THE POLAR BEAR

LIVES IN AN ENVIRONMENT WHERE

RETAINING HEAT IS VERY IMPORTANT.

SO LET'S TAKE A LOOK AT THE RATIO

OF SURFACE AREA TO VOLUME

FOR A TALL BUILDING LIKE THE CITIGROUP TOWER.

FOR SIMPLICITY LET'S SAY THAT THE SHAPE

IS THAT OF A SQUARE PRISM.

THE NET FOR A SQUARE PRISM SHOWS THAT THE SURFACE AREA

IS MADE UP OF TWO SQUARES AND FOUR RECTANGLES.

SUPPOSE WE LABEL THE SQUARE SIDES X.

RATHER THAN USING A DIFFERENT VARIABLE

FOR THE RECTANGULAR SIDE LETS DEFINE

THE LENGTH OF THE PRISM AS CX FOR SOME NUMBER C.

IN OTHER WORDS THE LENGTH OF THE SQUARE PRISM

IS SOME MULTIPLE OF THE SQUARE SIDE.

THIS MEANS THAT THE SURFACE AREA OF THE TOWER

BECOMES 2X SQUARED PLUS 4CX SQUARED.

THE VOLUME OF THE RECTANGULAR PRISM

IS THE AREA OF THE SQUARE BASE

TIMES THE LENGTH OR CX CUBED.

LET'S USE THE NSPIRE TO EXPLORE THE RATIO

OF SURFACE AREA TO VOLUME FOR A SQUARE PRISM.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

BEFORE INPUTTING A GRAPH CREATE A SLIDER.

THIS WILL BE USED TO VARY THE VALUES OF THE CONSTANT C

FROM OUR SURFACE AREA TO VOLUME RATIO.

PRESS MENU AND UNDER ACTIONS SELECT INSERT SLIDER.

SINCE THE SLIDER LABEL IS HIGHLIGHTED

CHANGE IT TO C AND PRESS ENTER.

NEXT PRESS THE TAB KEY TWICE TO GET TO

THE f1 FUNCTION ENTRY LINE.

INPUT THIS EXPRESSION.

PRESS ENTER.

THIS IS THE GRAPH OF A RATIONAL FUNCTION.

AS X APPROACHES ZERO THE GRAPH APPROACHES INFINITY.

SINCE X IS THE VALUE FOR THE SIDE LENGTH

OF THE SQUARE THEN THE GRAPH IS UNDEFINED

FOR X EQUALS ZERO.

WE'RE ONLY INTERESTED IN THE BEHAVIOR

OF THIS GRAPH IN THE FIRST QUADRANT.

PRESS MENU AND UNDER WINDOW/ZOOM

CHANGE THE SETTINGS TO THESE:

XMIN=0. XMAX=50, YMIN=0, YMAX=5.

PRESS OKAY AFTER YOU HAVE MADE THESE CHANGES.

THE VALUE OF C IS THE MULTIPLE

OF THE SQUARE SIDE LENGTH

THAT MAKES UP THE LENGTH OF THE SQUARE PRISM.

BY DEFAULT C EQUALS 5.

SO THIS GRAPH IS THE RATIO OF SURFACE AREA TO VOLUME

FOR A BUILDING THAT IS FIVE TIMES TALLER

THAN IT IS WIDE.

FOR SMALL VALUES OF X

THE RATIO OF SURFACE AREA TO VOLUME IS HIGH

MEANING THAT THE BUILDING LOSES HEAT.

BUT AS X INCREASES IN VALUE...

IN OTHER WORDS AS A BUILDING GETS WIDER AND TALLER

THE RATIO GOES DOWN AND IT DOES SO RATHER DRAMATICALLY.

TALL BUILDINGS THAT MAINTAIN A 5 TO 1 RATIO

FOR THE SIDE LENGTH AND HEIGHT

CAN ACHIEVE A GOOD RATIO WHERE HEAT IS RETAINED.

BUT LET'S LOOK AT VALUES THAT ARE CLOSER

TO THOSE FOUND WITH SKYSCRAPERS.

THE CITIGROUP TOWER IS ROUGHLY SIX TIMES

THE HEIGHT OF THE SQUARE BASE.

THE SQUARE BASE IS ROUGHLY 46 METERS IN LENGTH.

FIRST MOVE THE POINTER ABOVE THE SLIDER.

PRESS AND HOLD THE CLICK KEY OVER THE SLIDING ARROW

UNTIL THE OPEN HAND TURNS TO A GRASPING HAND.

USE THE RIGHT ARROW KEY TO MOVE THE SLIDER

TO VALUE C EQUALS 6.

PRESS ESCAPE.

NEXT PRESS MENU AND UNDER TRACE SELECT GRAPH TRACE.

MOVE THE POINTER TO X EQUALS 46

OR SIMPLY INPUT 46 AND PRESS ENTER.

YOU'LL SEE THAT THE VALUE OF SURFACE AREA OVER VOLUME

FOR THE CITIGROUP BUILDING IS QUITE SMALL.

SO THE CITIGROUP BUILDING DOES HAVE

A GOOD SURFACE AREA TO VOLUME RATIO.

THE SURFACE AREA TO VOLUME RATIO FOR TALL BUILDINGS

IS SOMETHING AN ARCHITECT NEEDS TO BE AWARE OF.

KEEPING A SKYSCRAPER WARM IN WINTER AND COOL IN SUMMER

REQUIRES A GREAT DEAL OF ENERGY.

AS A RESULT, AN AREA OF CONCERN FOR ARCHITECTS

IS THE ENERGY EFFICIENCY OF TALL BUILDINGS.

WHILE TALL, SLEEK BUILDINGS HAVE VISUAL APPEAL,

BALANCING THE ARTISTIC NEEDS WITH THE ECONOMIC ONES

IS IMPORTANT.

NEWER BUILDINGS HAVE TAKEN THESE CONSIDERATIONS

INTO ACCOUNT AND HAVE RESULTED IN INCREASING

INTEREST IN SO-CALLED GREEN ARCHITECTURE.

MODIFICATIONS TO EXISTING BUILDINGS

LIKE THE CITIGROUP TOWER

ARE OFTEN TO MAKE THEM MORE ENERGY EFFICIENT.