- WE ARE GOING TO LOOK AT ONE OF THE MOST POWERFUL
USES OF MATHEMATICS, AND THAT IS MATHEMATICAL MODELING.
ESSENTIALLY WHAT WE'RE GOING TO DO IS,
GIVEN A MATHEMATICAL EQUATION THAT MODELS A SITUATION,
WE WILL BE ABLE TO MAKE PREDICTIONS ABOUT THE FUTURE.
LET'S TAKE A LOOK AT HOW THIS PROCESS NORMALLY WORKS.
NORMALLY, WHAT HAPPENS IS YOU HAVE A REAL LIFE
PROBLEM OR SITUATION.
AND THEN FROM THAT PROBLEM OR SITUATION,
YOU'RE ABLE TO GATHER A BUNCH OF DATA.
AND THEN FROM THAT DATA,
YOU USE MATHEMATICS TO FIND A MATHEMATICAL MODEL
THAT RESEMBLES THAT DATA.
ONCE YOU HAVE THE MATHEMATICAL MODEL,
YOU'RE ABLE TO MAKE PREDICTIONS ABOUT THE FUTURE.
NOW, TRADITIONALLY, IT IS A FOUR-STEP PROCESS.
BUT IN THE SITUATION FOR TODAY,
WE'RE GOING TO BE GIVEN THE MATHEMATICAL MODEL
THAT MODELS THE DATA,
AND THEN WE'RE GOING TO MAKE OUR PREDICTIONS.
LATER ON, WE WILL ADDRESS THESE OTHER TWO ISSUES.
LET'S TAKE A LOOK AT AN EXAMPLE.
THE GRAPH BELOW SHOWS A NUMBER OF DIGITAL PHOTOS
PRINTED AT HOME FROM THE YEAR 2000 TO 2006.
IT'S PROBABLY HARD FOR YOU TO READ THIS,
BUT YOU CAN SEE THAT THE TREND IS THAT PEOPLE ARE PRINTING
MORE AND MORE DIGITAL PHOTOS AT HOME.
HERE'S THE PROBLEM.
USE THE MODEL H = 0.7T + 0.3
WHERE T IS THE NUMBER OF YEARS AFTER 2000
AND H IS THE NUMBER OF DIGITAL PHOTOS
PRINTED AT HOME IN BILLIONS.
PREDICT THE NUMBER OF DIGITAL PHOTOS PRINTED AT HOME
IN THE YEAR 2008.
LET'S WRITE THIS A LITTLE LARGER, SO WE CAN SEE IT.
H = (7/10)T + 3/10.
OKAY, NOW, T IS NOT EQUAL TO 2008
BECAUSE T IS THE NUMBER OF YEARS AFTER 2000.
SO FOR THE YEAR 2000, THAT WOULD BE T = 0.
SO IF I WANT THE YEAR 2008-- OR WE WANT THE YEAR 2008,
IT'S PRETTY EASY TO SEE IN THIS CASE THAT T = 8.
AN EASY WAY TO FIND THIS,
IF IT'S CHALLENGING, IS TAKE THE YEAR THAT WE WANT
AND SUBTRACT THE BASE YEAR OF 2000.
'COURSE THAT WOULD GIVE US EIGHT.
WE NEED TO SUBSTITUTE 8 IN FOR T.
OKAY, LET'S GO AHEAD AND EVALUATE THIS
ON OUR GRAPHING CALCULATORS.
SO I HAVE 0.7 x 8 - 0.3, AND WE GET 5.9.
H = 5.9.
SO WHAT'S OUR ANSWER?
WELL, BEFORE WE FIGURE THAT OUT, LET'S SEE IF THAT MAKES SENSE.
AGAIN, THIS IS FOR WHAT WE'RE SAYING FOR THE YEAR 2008.
IF WE LOOK BACK AT THE GRAPH, THE PROJECTION IS IN 2006,
IT'LL BE 4.5.
SO IF THE TREND IS UPWARD, 5.9 IS A REASONABLE ANSWER.
HOWEVER, IF YOU LOOK BACK AT THE PROBLEM, H IS IN BILLIONS.
OKAY, SO OUR FINAL ANSWER WOULD BE 5.9 BILLION PHOTOS.
OKAY, IF WE WANT TO CONVERT THIS TO EXPANDED NOTATION,
WE WOULD TAKE THE 5.9 AND MULTIPLY IT BY 1 BILLION.
NOW, IF YOU'RE NOT SURE HOW MANY ZEROES, JUST TAKE A MOMENT.
THERE'S A THOUSAND, AND THAT WOULD BE A MILLION.
SO WE NEED THREE MORE, SO THAT WOULD BE A BILLION.
SO IN EXPANDED NOTATION, THIS WOULD BE 5 BILLION,
900 MILLION PHOTOS.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
USE THE MODEL N = 38T + 829
TO PREDICT THE NUMBER OF NURSES WORKING FULL-TIME
WHERE T IS THE NUMBER OF YEARS AFTER 1980
AND N IS THE NUMBER OF NURSES IN THE THOUSANDS.
PREDICT THE NUMBER OF NURSES THAT WILL WORK FULL-TIME
IN 2012.
OKAY, SO WE HAVE A SIMILAR SITUATION HERE.
WE NEED TO FIGURE OUT,
WHAT ARE WE GOING TO USE FOR THE VALUE OF T?
WELL, T IS THE NUMBER OF YEARS AFTER 1980,
SO I'LL TAKE THE YEAR THAT WE WANT
AND I'LL SUBTRACT OUR BASE YEAR.
AND WE CAN SEE THAT'S GOING TO GIVE US 32.
WE'RE GOING TO REPLACE T WITH THE VALUE OF 32.
AND LET'S GO TO OUR CALCULATOR TO SIMPLIFY THIS.
38 x 32 + 829--2,045.
NOW, AGAIN, A COMMON MISTAKE IS TO SAY
THE ANSWER IS 2,045 NURSES.
BUT REMEMBER THE NURSES IS IN THOUSANDS.
SO TO FIND THE TRUE NUMBER OF NURSES,
WE WOULD HAVE TO MULTIPLY THIS BY 1,000.
AND THIS WOULD BE THE NUMBER OF NURSES WORKING FULL-TIME.
OKAY, I HOPE THAT HELPS EXPLAIN
HOW YOU WOULD USE A MODEL TO MAKE A PREDICTION.
HAVE A GOOD DAY.