Get the Math

female announcer: GET THE MATH 2.0 IS FUNDED BY THE NEXT GENERATION LEARNING CHALLENGES... WITH ADDITIONAL SUPPORT FROM THE MOODY'S FOUNDATION.

[hip-hop music] - UH, YEAH, UH I WANNA MASTER THE TASK OF EVERYDAY MATH KIDS, IT'S A CRITICAL CRAFT, LISTEN IN CLASS NOW YOU CAN ADD, SUBTRACT, MULTIPLY, DIVIDE THE STACKS GOTTA KEEP YOUR NICKELS AND DIMES INTACT NOW YOU CAN COUNT THE AMOUNT OF LYRICAL BARS SO YOU CAN STAY UNDER THE SKY DIVIDING THE STARS CALCULATE YOUR SALES TAX AND KNOW WHAT THEY CHARGE SO YOU CAN KEEP TRACK OF THE DOUGH THEY SWIPE ON YOUR CARD YEAH, THEY CLAIM IT'S DIFFICULT BUT IT ISN'T THAT HARD IF YOU'RE SMART WITH THE DIGITS YOU MIGHT GET YOU A CAR A HOUSE, A BOAT, IF YOU'RE LUCKY, A PLANE SO WHEN YOU'RE PASSING DIFFERENT TIME ZONES THE WEATHER WILL CHANGE STAY ON POINT LIKE STALACTITES NEED THE MATH TO MAKE THE MEGA MUSIC AND GET YOUR CASH RIGHT IF YOU GOT A WORD PROBLEM AND THINGS LOOK BAD GET THE MATH IF YOU WONDER ABOUT THE NUMBERS, DON'T BE SAD GET THE MATH WHEN YOU CAN'T GET THE SOLUTION, DON'T GET MAD GET THE MATH FIND YOUR ANGLE WITH EXAMPLES, THAT'S THE PLAN [echoing] GET THE MATH - LOTS OF PEOPLE USE MATH IN THEIR WORK.

SUE TORRES USES MATH IN HER RESTAURANT.

- FIRE UP TWO BREAD.

MATH PLAYS A MAJOR ROLE IN FIGURING OUT HOW TO MAKE A RESTAURANT PROFITABLE.

- ELTON BRAND USES MATH IN BASKETBALL.

- I DON'T CONSTANTLY THINK ABOUT THE MATH BEHIND IT WHEN I'M PLAYING A GAME, BUT WHEN I WAS YOUNGER, UNDERSTANDING MATH HELPED ME IMPROVE MY PERFORMANCE.

- JEREMY CHERNICK USES MATH IN SPECIAL EFFECTS.

- THERE'S MATH IN EVERY EFFECT THAT WE CREATE HERE IN THE SHOP.

- THEY EACH POSE A CHALLENGE LIKE ONE THEY'RE TACKLED ON THE JOB.

- AFTER YOU GIVE EACH PROBLEM A SHOT, YOU CAN SEE HOW WE TRIED TO SOLVE IT USING ALGEBRA AND IF WE WERE ABLE TO GET THE MATH.

- MY NAME IS SUE TORRES, AND I AM THE CHEF AND OWNER OF SUENOS RESTAURANT IN CHELSEA, NEW YORK, WHICH IS A MEXICAN RESTAURANT.

IF YOU WATCH COOKING SHOWS, YOU MAY HAVE SEEN ME ON IRON CHEF AMERICA BATTLING BOBBY FLAY, OR JUDGING ON THE SHOW CHOPPED.

MOST RECENTLY, I WAS COMPETING ON TOP CHEF MASTERS.

I'VE BEEN INTERESTED IN COOKING SINCE I WAS A KID.

GROWING UP, I LEARNED HOW TO COOK THE BEST ITALIAN AND PUERTO RICAN DISHES FROM WATCHING THE INCREDIBLE WOMEN COOKS IN MY FAMILY.

IN 11TH AND 12TH GRADE, I WENT TO A VOCATIONAL CULINARY SCHOOL WHERE I WAS ABLE TO COOK IN A PROFESSIONAL KITCHEN, AND THEN WHEN I GRADUATED, I WENT TO THE CULINARY INSTITUTE OF AMERICA.

I'M HALF ITALIAN AND HALF PUERTO RICAN, BUT MY HEART IS MEXICAN.

I DECIDED TO OPEN UP MY OWN MEXICAN RESTAURANT BECAUSE I WANTED TO DO THINGS TRUE AND AUTHENTIC TO MEXICO, AND AS AN OWNER, YOU HAVE THE OPTION TO DO THINGS EXACTLY HOW YOU WANT IT.

DON'T BE FOOLED BY COOKING SHOWS.

A CAREER AS A CHEF, IN GENERAL, IS NOT GLAMOROUS.

FIRE UP TWO BREAD.

IT REQUIRES HARD WORK, DETERMINATION.

YOU'RE CONSTANTLY OVERCOMING DAY TO DAY.

HELLO.

ONE OF THE BIGGEST CHALLENGES IS KEEPING THE PRICES ON THE MENU AFFORDABLE WHILE THE COSTS OF INGREDIENTS RISE, SO MATH PLAYS A MAJOR ROLE IN FIGURING OUT HOW TO MAKE A RESTAURANT PROFITABLE.

GUACAMOLE IS ONE OF THE MOST POPULAR APPETIZERS HERE AT SUENOS, BUT PRICING IS TRICKY BECAUSE THE COST OF AVOCADO-- THE MAIN INGREDIENT-- CHANGES A LOT.

YOUR CHALLENGE IS TO LOOK AT AVOCADO PRICES FROM THE PAST THREE YEARS TO PREDICT WHAT AVOCADOS MIGHT COST IN THE NEXT 12 MONTHS.

THEN, USING THE PREDICTION, RECOMMEND A MENU PRICE FOR GUACAMOLE FOR THE NEXT YEAR AT SUENOS.

THE TABLE AND GRAPH SHOW THE COST OF A CASE OF 48 AVOCADOS OVER THE PAST THREE YEARS.

YOU'LL WANT TO LOOK FOR A TREND AND MAKE A PREDICTION ABOUT THE COST OF AVOCADOS FOR THE NEXT 12 MONTHS.

THEN, BASED ON YOUR PREDICTION, YOU CAN USE MY RULE OF THUMB FOR PRICING.

GENERALLY SPEAKING, THE MENU PRICE IS ABOUT FOUR TIMES THE COST OF INGREDIENTS.

FOR GUACAMOLE, EACH PORTION INCLUDES THE COST OF ONE AVOCADO, PLUS ABOUT 40 FOR THE ADDITIONAL INGREDIENTS.

COME UP WITH ONE PRICE THAT I COULD CHARGE FOR ALL OF THE NEXT YEAR BASED ON YOUR PREDICTIONS AND MY RULE OF THUMB.

GOOD LUCK, AND GET STARTED.

- SO I THINK WE CAN START OFF BY DRAWING A BEST FIT LINE TO FIND THE AVERAGE RATE OF CHANGE BY WHICH THE COST OF THE CASE IS INCREASING.

- OKAY.

- WE WANT TO TRY AND GET, LIKE, THE SAME AMOUNT OF POINTS ON BOTH SIDES OF THE LINE.

- WE USED A LINE OF BEST FIT BECAUSE WHEN YOU HAVE A VARIETY OF PRICES, A LINE OF BEST FIT CAN HELP YOU FIND A TREND.

IT BASICALLY GOES THROUGH THESE TWO POINTS.

- SO IT WAS THE NINTH MONTH, AND AVOCADOS WERE ABOUT $36 FOR A CRATE.

- YES, AND THEN WE HAVE JUNE OF 2011.

- AVOCADOS WERE ABOUT $45 A CRATE.

- THE FIRST POINT IS AN "X" VALUE OF 3, AND ITS "Y" VALUE IS 28.

- OKAY.

- AND THEN THE LAST ONE THAT WE USED IS...

IT'S THE SECOND TO LAST ONE.

ITS "X" VALUE IS 33 AND ITS "Y" VALUE IS 52.

SO NOW THAT WE HAVE OUR POINTS, WE CAN FIND THE SLOPE BY DIVIDING THE DIFFERENCE OF THE COST BY THE DIFFERENCE OF THE MONTHS, WHICH IS 80.

- SO THE VALUES ARE INCREASING BY 0.8, SO NOW THAT WE'RE WORKING WITH A BEST FIT LINE, WE CAN SAY THAT AT 33 MONTHS, 52 WAS OUR PRICE, AND AT 34 MONTHS, IT'S GOING TO BE 52 PLUS 0.8.

- OKAY, I'M JUST GONNA MAKE A SMALL TABLE.

- OH, OKAY.

BECAUSE WE KNEW THAT THE PRICE OF AVOCADOS INCREASED AT A RATE OF 80 PER MONTH, WE SIMPLY ADDED 80 TO EVERY MONTH UNTIL WE REACHED THE END OF THE NEXT YEAR.

SO IF WE DIVIDE THAT BY 48 AVOCADOS, WE CAN GET THE COST PER AVOCADO.

$1.32.

- OKAY.

- SO BECAUSE WE'RE FINDING AVERAGE COST OF AVOCADOS FOR THAT YEAR, DO YOU WANT TO TAKE THE FIRST ONE OF THE MONTH?

- OKAY.

- SO WHAT DID YOU GET FOR, I GUESS, JANUARY OF 2012?

- I GOT 55.2.

- OKAY.

AND THEN WE'LL DIVIDE THAT BY 48 TO SEE THE AVERAGE COST OF AN AVOCADO IN JANUARY, RIGHT?

- RIGHT.

I GOT $1.15.

- I GOT THE SAME THING.

- OKAY.

SO NOW THAT WE HAVE THE TWO POINTS, WE CAN GO AND FIND THE DIFFERENCE BETWEEN THEM AND THEN DIVIDE IT BY 2 TO FIND A GENERAL IDEA OF WHERE THE AVERAGE IS GOING TO BE, RIGHT?

- OKAY.

1.24.

- YEAH.

both: $1.24.

- SO THAT'S AN AVERAGE COST OF ONE AVOCADO IN THE YEAR 2012.

- SO NOW THAT WE HAVE THAT, WE NEED TO APPLY THE RULES THAT SHE GAVE US BEFORE, RIGHT?

- OKAY, SO WE CAN SAY THAT HER MENU PRICE IS ABOUT 4 TIMES THE COST OF INGREDIENTS, AND WE KNOW THAT THE COST OF INGREDIENTS IS GONNA BE THE COST OF THE AVOCADO, WHICH IS $1.24, PLUS THE COST OF 40 ADDITIONAL INGREDIENTS.

- OKAY, SO FOR ONE PORTION OF GUACAMOLE, WE HAVE IT AT $6.56.

- I THINK WE SHOULD ROUND IT TO THE NEAREST DOLLAR, BECAUSE I DON'T THINK ANYONE LOOKING AT A MENU WOULD WANT TO SAY, "OH, WE HAVE TO PAY $6.56 FOR THIS."

I THINK $7 WOULD WORK, RIGHT?

- OKAY, THAT SOUNDS ABOUT RIGHT.

- SO LET'S FIND THE SLOPE.

WE SHOULD DO RISE OVER RUN, SO THE DIFFERENCE IN THE COST FOR THE CASE OVER THE DIFFERENCE OF MONTHS.

SO IT WOULD BE 45 MINUS 36 OVER 30 MINUS 9.

AND GET 9 OVER 21, RIGHT?

- YUP.

- SO THE SLOPE WOULD BE 3 OVER 7.

- OVER 7.

- LET'S FIND THE EQUATION OF THE LINE.

- WE HAVE THE SLOPE AND WE HAVE A POINT, WHY DON'T WE USE THE POINT-SLOPE FORM?

- THE POINT-SLOPE FORM IS "Y" MINUS Y1 EQUALS "M," OR THE SLOPE, TIMES "X" MINUS X1.

AT THE NINTH MONTH, THE COST OF AN AVOCADO CASE WAS $36.

AND WE NOW KNOW THAT THE SLOPE IS 3/7, SO THE EQUATION FOR THAT WOULD BE "Y" MINUS 36 EQUALS 3/7 TIMES "X" MINUS 9.

BUT WE WANT TO FIND OUT THE COST FOR 2012.

- SO WE CAN USE JUNE, WHICH IS IN THE MIDDLE, WHICH IS THE 42ND MONTH.

- ACCORDING TO OUR BEST-FIT LINE, THE PRICES OF AVOCADOS WOULD BE LOWER IN THE BEGINNING OF THE YEAR AND HIGHER LATER ON IN THE YEAR, SO WE CHOSE THE MIDPOINT IN ORDER TO FIND THE AVERAGE.

SO THEN WE NEED TO FIND A PRICE FOR THE MENU.

- SO WE WOULD DIVIDE THAT BY-- - 48.

SO ONE AVOCADO ON ITS OWN IS ABOUT $1.04.

SO THEN, PLUS THE OTHER INGREDIENTS, IT COMES TO A $1.04 PLUS 40.

- WHICH WOULD BE $1.44.

TIMES 4 IS $5.77.

- DO YOU THINK THAT THAT WOULD BE A GOOD PRICE TO PUT ON THE MENU?

- LET'S ROUND THAT UP TO $6.

- $6?

OKAY, THAT SOUNDS GOOD TO ME.

- SO WHAT DID YOU GUYS COME UP WITH?

- WE USED A LINE OF BEST FIT THAT CAME UP WITH A RATE OF CHANGE OF 80 PER MONTH.

- SO WE RECOMMEND CHARGING $7 PER PORTION.

- WE ALSO MADE A LINE OF BEST FIT, BUT WE FOUND THAT A CRATE OF AVOCADOS WOULD INCREASE BY 43 PER MONTH.

- SO WE RECOMMEND $6.

- SO WHY DO YOU THINK YOU HAVE TWO DIFFERENT ANSWERS?

- BOTH GROUPS USED LINES OF BEST FIT, BUT WE HAD DIFFERING RATES OF CHANGE BECAUSE WE MUST HAVE USED DIFFERENT POINTS.

- NICE WORK.

OF COURSE, MENU PRICING ISN'T AN EXACT SCIENCE.

I ALSO CONSIDER WHAT PEOPLE WOULD BE WILLING TO PAY, HOW POPULAR THE GUACAMOLE IS ON THE MENU, AND WHAT OTHER RESTAURANTS ARE CHARGING.

SO TRY OUR GUACAMOLE AT SUENOS.

DIG IN.

- I ACTUALLY REALLY LIKED THE GUACAMOLE.

IT WAS MY FIRST TIME EVER HAVING IT, SO I REALLY LIKED IT A LOT, AND I PLAN ON EATING A LOT MORE IN THE FUTURE.

- I'M ELTON BRAND.

I'M A POWER FORWARD IN THE NBA.

AND I'M HERE IN MY HOMETOWN OF PEEKSKILL, NEW YORK, VISITING MY OLD HIGH SCHOOL.

I STARTED PLAYING BASKETBALL WHEN I WAS ABOUT FIVE, SIX, YEARS OLD, AND I WAS REALLY, REALLY BAD BUT LEARNED THE GAME AND, YOU KNOW, KEPT WORKING AT IT, AND GOT BETTER AND BETTER AS I GOT OLDER.

I STARTED OUT HIGH SCHOOL ON THE JUNIOR VARSITY TEAM.

THEN THE COACH FROM THE VARSITY TEAM BROUGHT ME UP TO THAT LEVEL.

THE FOLLOWING TWO YEARS, WE WON THE NEW YORK STATE CHAMPIONSHIP, WHICH PUT ME ON THE NATIONAL SPOTLIGHT, AND MY TEAM.

- AGAIN BRAND!

OH... - SO AFTER HIGH SCHOOL, I ACCEPTED A SCHOLARSHIP TO DUKE UNIVERSITY, GOT TO COLLEGE, AND PLAYED REALLY, REALLY WELL AND BECAME THE NUMBER ONE PICK OUT OF ALL THE COLLEGIATE PLAYERS AND GOT DRAFTED BY THE CHICAGO BULLS NUMBER ONE OVERALL.

AND IT WAS A DREAM COME TRUE.

AS ANYONE THAT FOLLOWS SPORTS STATISTICS KNOWS, THERE'S A LOT OF MATH IN BASKETBALL.

FREE THROW SHOOTING IS A SKILL THAT EVERY BASKETBALL PLAYER HAS TO WORK ON.

I DON'T CONSTANTLY THINK ABOUT THE MATH BEHIND IT WHEN I'M PLAYING A GAME, BUT WHEN I WAS YOUNGER, UNDERSTANDING MATH HELPED ME IMPROVE MY PERFORMANCE.

THREE KEY VARIABLES CAN INFLUENCE YOUR SHOT.

FIRST IS THE RELEASE HEIGHT.

SECOND IS THE BALL'S INITIAL VERTICAL VELOCITY, A RATE THAT INCLUDES THE ANGLE AND SPEED OF THE BALL AS IT LEAVES YOUR HAND.

AND THE THIRD IS GRAVITY, WHICH MAKES A BASKETBALL ACCELERATE AS IT FALLS.

YOU CAN USE THESE VARIABLES TO FIGURE OUT THE HEIGHT OF THE BALL AT ANY GIVEN TIME, AND KNOWING THAT, YOU CAN TELL IF YOU'LL MAKE THE SHOT.

WHEN I SHOOT A FREE THROW, MY GOAL IS TO MAXIMIZE THE HEIGHT OF THE BALL SO IT HAS THE BEST CHANCE OF GOING IN.

YOUR CHALLENGE IS TO USE THE THREE KEY VARIABLES AND MY STATS TO FIGURE OUT THE MAXIMUM HEIGHT THE BALL REACHES ON ITS WAY INTO THE BASKET.

AT RELEASE, MY AVERAGE INITIAL VERTICAL VELOCITY IS 24 FEET PER SECOND.

MY AVERAGE RELEASE HEIGHT IS 7 FEET.

THE FREE THROW LINE IS 15 FEET FROM THE BACKBOARD.

THE HEIGHT OF THE BASKETBALL HOOP IS 10 FEET OFF THE FLOOR, AND THE RIM IS 18 INCHES IN DIAMETER.

AND FINALLY, YOU CAN FIND THE HEIGHT OF THE BALL AT EACH POINT IN TIME BY COMBINING THE THREE KEY VARIABLES: THE ACCELERATION OF GRAVITY, WHICH IS NEGATIVE 32T SQUARED, BUT IN THIS CASE, YOU'D USE ONLY HALF FOR THE DOWNWARD PULL, OR NEGATIVE 16T SQUARED; THE INITIAL VERTICAL VELOCITY, WHICH YOU'D MULTIPLY BY TIME TO GET THE DISTANCE TRAVELED; AND THE RELEASE HEIGHT.

ALL RIGHT, GET TO WORK.

- SO THE ACCELERATION OF GRAVITY WOULD BE NEGATIVE 16T SQUARED.

- THE INITIAL VERTICAL VELOCITY IS 24 FEET PER SECOND TIMES "T." - MM-HMM.

- SO THIS KIND OF LOOKS LIKE IT COULD BE, LIKE, A QUADRATIC EQUATION, RIGHT?

- YEAH.

- WHERE "T" WOULD BE ON THE "X" AXIS, AND FEET WOULD BE ON THE "Y" AXIS, RIGHT?

- YEAH, SO WHEN WE WRITE THIS DOWN AS AN EQUATION, WE COULD USE NEGATIVE 16T SQUARED PLUS 24T PLUS 7.

- SO WE HAVE "Y" EQUALS NEGATIVE 16T SQUARED PLUS 24T.

AND THEN WE'RE GONNA ADD THE RELEASE HEIGHT, WHICH IS 7 FEET.

- EXACTLY.

SO NOW WE HAVE OUR EQUATION.

- LET'S GRAPH IT FIRST, JUST TO SEE HOW IT WOULD LOOK LIKE ON THE GRAPHING CALCULATOR.

- OKAY.

SKETCHING A GRAPH IS A GOOD IDEA JUST BEFORE USING YOUR CALCULATOR BECAUSE IT WILL GIVE YOU A GENERAL IDEA OF WHAT YOUR GRAPH SHOULD LOOK LIKE, IN CASE YOU MAKE ANY MISTAKES ON YOUR CALCULATOR.

- OUR "X" AXIS IS GONNA BE TIME.

- RIGHT.

- AND OUR "Y" AXIS IS GONNA BE DISTANCE, WITH TIME BEING IN SECONDS AND DISTANCE BEING IN FEET.

SO SINCE HE'S 7 FEET OFF THE GROUND, WE CAN JUST HAVE OUR FIRST POINT BE (0,7).

AND THEN, THE BALL'S GONNA MAKE SORT OF THIS UPWARD TRAJECTORY, RIGHT?

- RIGHT.

IT'S GONNA BE AN UPWARD ARCH.

- YEAH, AND THEN THAT HIGHEST POINT IS GONNA BE WHERE THE BALL REACHES ITS MAXIMUM HEIGHT.

- ALL RIGHT.

- AND IT'S GONNA COME BACK DOWN INTO THE BASKET.

BUT WE HAVE TO NOTE THAT THE BASKET IS ACTUALLY 10 FEET OFF THE GROUND, RIGHT?

- IT'S GONNA BE (T,10)?

- YEAH.

- I HAVE A GENERAL IDEA-- - OF WHERE OUR MAXIMUM POINT IS GONNA BE.

- RIGHT.

- OKAY.

SO WE CAN JUST CHECK IT OUT ON THE GRAPHING CALCULATOR.

SO NOW WE'RE LOOKING FOR THE HIGHEST POINT.

LET'S SET THE LOWER AND UPPER BOUNDS.

- OKAY, WE GO TO THE MAXIMUM.

- YEAH, WE WANT IT TO BE AS CLOSE TO OUR MAXIMUM AS POSSIBLE, BUT IT DOESN'T REALLY MATTER WHERE WE PICK IT, JUST AS LONG AS IT'S IN THE RANGE OF THE LEFT- AND THE RIGHT-HAND SIDE BECAUSE WE HAVE A SYMMETRICAL PARABOLA.

WHAT'D YOU GET?

- OKAY, SO FOR MY "X," I GOT 0.7499, WHICH IS JUST ABOUT 0.75, AND FOR MY "Y" I GOT EXACTLY 16.

- I GOT EXACTLY 16 FOR "Y" TOO, AND I GOT 0.7500, SO WE HAVE ABOUT THE SAME THING.

- RIGHT, IT ROUNDS UP TO 0.75 FOR OUR TIME, WHICH IS GONNA BE WHERE THE BALL REACHES ITS MAXIMUM HEIGHT, WHICH IS GOING TO BE 16 FEET OFF THE GROUND.

- EXACTLY, SO WE HAVE OUR ANSWER.

- OKAY , WELL, THE END RESULT IS GOING TO BE 10.

- RIGHT, THE HEIGHT OF THE BASKETBALL HOOP.

SO WE'RE SETTING "Y" EQUAL TO 10 FEET.

NOW WE HAVE TO FIND OUT WHAT "T" EQUALS, SO WE HAVE TO MOVE EVERYTHING TO ONE SIDE, SO WE GET 0 EQUALS NEGATIVE 16T SQUARED PLUS 24T MINUS 3, 'CAUSE WE'RE SUBTRACTING 10 FROM EACH SIDE.

SO "X" EQUALS NEGATIVE "B" PLUS OR MINUS THE SQUARE ROOT OF "B" SQUARED MINUS 4AC, ALL OVER 2A.

SO IF WE PLUG IN THE VALUES FOR "A," "B," AND "C," WHEN WE CONSIDER 0 IS EQUAL TO AX SQUARED PLUS BX PLUS "C," THEN WE GET NEGATIVE 24 PLUS OR MINUS THE SQUARE ROOT OF... - 24 SQUARED.

- RIGHT.

- MINUS 4 TIMES-- - NEGATIVE 16 TIMES NEGATIVE 3, AND THEN ALL OVER NEGATIVE 32.

THEN WE GET TWO VALUES OF "X," OR TWO DIFFERENT TIMES WHERE THE BALL REACHES THE HEIGHT OF 10 FEET, SO 3 PLUS OR MINUS RADICAL 6 OVER 4.

- THE 3 PLUS THE SQUARE ROOT OF 6 DIVIDED BY 4 IS 1.36.

3 MINUS THE SQUARE ROOT OF-- - RADICAL 6 DIVIDED BY 4-- - IS 0.14.

- 0.14, OKAY.

SO IF WE WERE TO IMAGINE THIS LIKE A PARABOLA, LIKE IT IS, THEN WE HAVE SYMMETRY.

- YEAH, WE HAVE BOTH THE POINTS AT 10.

SO THOSE WOULD BE THE "X" VALUES OF 10.

- RIGHT.

SO THEN, IF WE FIND THE AVERAGE OF THESE TWO "X" VALUES, THEN WE CAN FIND THE "X" VALUE OF THE VERTEX OF THE PARABOLA, WHICH WOULD GIVE US THE MAXIMUM POINT OF THE PARABOLA, THUS GIVING US THE MAXIMUM POINT OF THE BALL.

- SO THAT'S 1.36-- - RIGHT.

- PLUS 0.14-- - RIGHT.

- DIVIDED BY 2-- - TO FIND THE AVERAGE.

- WHICH IS 0.75.

- OKAY, SO 0.75.

SO NOW WE HAVE THE "X" VALUE FOR THE VERTEX OF THE PARABOLA, SO IF WE PLUG THAT INTO THE EQUATION OF THE PARABOLA, THEN WE SHOULD GET THE "Y" VALUE, OR THE MAXIMUM HEIGHT OF THE BALL.

SO NEGATIVE 16 TIMES 0.75 SQUARED.

- 18 PLUS 7 MINUS 9 IS 16.

- SO AT 3/4 OF A SECOND, THE BALL REACHES ITS MAXIMUM HEIGHT OF 16 FEET.

- MM-HMM.

- SO HOW'D IT GO?

WHAT ANSWER DID YOU GET?

- WELL, WE GOT THE MAXIMUM HEIGHT OF 16 FEET, AND WE GOT THAT BY USING THE ACCELERATION OF GRAVITY, THE INITIAL VERTICAL VELOCITY, AND THE RELEASE HEIGHT, AND COMBINING THAT TO MAKE THE QUADRATIC EQUATION.

- BY USING THE QUADRATIC FORMULA, WE FOUND TWO TIME VALUES WHERE THE BALL REACHED A HEIGHT OF 10 FEET.

BY AVERAGING THOSE TWO TIME VALUES, WE FOUND THE TIME WHERE THE BALL WAS AT ITS MAXIMUM HEIGHT, WHICH WE THEN FOUND TO BE 16 FEET.

- WE ALSO CAME UP WITH THE ANSWER OF 16 FEET FOR OUR MAXIMUM HEIGHT, BUT WE SOLVED IT A LITTLE BIT DIFFERENTLY.

FIRST, WE CAME UP WITH A GENERAL EQUATION, TO WHICH WE DREW A SKETCH.

- AND FROM THE SKETCH, WE USED A GRAPHING CALCULATOR TO CONFIRM THE RANGE IN WHICH THE MAXIMUM HEIGHT WOULD FALL.

- I THINK IT WAS INTERESTING THAT CARTER AND I SOLVED THE PROBLEM ALGEBRAICALLY, WHILE TIFFANY AND CALEN SOLVED IT GRAPHICALLY.

AND IT'S ONE OF THE THINGS THAT I LIKE MOST ABOUT MATH, IS THAT THERE'S ALWAYS AN ANSWER, BUT THERE ARE MULTIPLE WAYS TO ACHIEVE THAT ANSWER.

- UNDERSTANDING THE MATH INVOLVED CAN HELP IMPROVE YOUR PERFORMANCE, BUT A WHOLE LOT OF PRACTICE IS REALLY THE KEY.

IT'S MORE OF A PUSH UP THAN, LIKE, THROWING IT.

- OKAY.

- SO... - OKAY.

- YEAH.

BUT YOU'VE GOT TO USE THAT POWER IN YOUR LEGS--EXACTLY!

- OH!

- YES!

[laughs] YES!

[percussive rock music] - MY NAME IS JEREMY CHERNICK, AND I DESIGN SPECIAL EFFECTS FOR THEATER, ROCK SHOWS, TV, MOVIES, AND MUSIC VIDEOS.

I WORK HERE AT J&M SPECIAL EFFECTS, AND HERE WE DO PYROTECHNICS, CONFETTI, BREAKAWAY GLASS.

WE DO THE WEATHER, WHICH MEANS WE DO RAIN AND FOG AND SNOW.

I REALLY, REALLY LOVE WHAT I DO.

IT'S KIND OF SURPRISING THAT I'VE ENDED UP IN A CAREER THAT INVOLVES A LOT OF MATH.

I HAVE A LEARNING DISABILITY THAT MAKES IT VERY DIFFICULT FOR ME TO MEMORIZE ALL KINDS OF STUFF, ESPECIALLY THINGS WITHOUT ANY LOGIC TO IT, AND WHERE I WENT TO SCHOOL, MATH WAS TAUGHT IN A VERY TRADITIONAL WAY THAT REQUIRED MEMORIZING FORMULAS, SO I HAD A REALLY HARD TIME.

I WAS WAY MORE SUCCESSFUL DOING HANDS-ON STUFF, AND SO I STARTED GETTING INVOLVED IN THEATER.

AND I STARTED DOING A LOT OF PROPS FOR PLAYS.

IN MY PROFESSIONAL LIFE, THE PROP REQUESTS START GETTING MORE AND MORE ELABORATE, AND I ENDED UP CREATING A LOT OF SPECIAL EFFECTS.

THERE'S MATH IN EVERY EFFECT THAT WE CREATE HERE IN THE SHOP, BUT AT THIS POINT, I AM NOT THROWN BY THAT MATH ANYMORE BECAUSE, FOR ME, IT'S JUST PART OF SOLVING THE PROBLEM THAT IS THE EFFECT WE NEED TO CREATE.

HERE'S AN EFFECT THAT WE DID FOR A BAND CALLED THE FREELANCE WHALES FOR THEIR MUSIC VIDEO ENZYMES, AND IT'S AN EXPLODING FLOWER EFFECT.

FIRING IN THREE, TWO, ONE.

- [shrieks] - BECAUSE EXPLOSIONS HAPPEN SO QUICKLY-- BOOM, EXPLOSION, IT'S OVER-- THIS WAS SHOT AT SUPER HIGH SPEED.

IT WAS 2,000 FRAMES PER SECOND, AND WHEN YOU'RE SHOOTING NORMAL VIDEO, YOU'RE SHOOTING 24 TO 30 FRAMES PER SECOND, SO WE HAD TO SHOOT THIS SO FAST AND, LIKE, REALLY SLOW IT DOWN.

- AND ONE OF CHALLENGES WAS GETTING THE LIGHTING RIGHT.

AS A GENERAL RULE, THE FASTER YOUR SUBJECT IS MOVING, THE MORE LIGHT YOU NEED TO THROW ON IT.

- THIS IS ONE OF THE FIRST SHOTS WE TOOK.

WHAT DO YOU THINK OF THE LIGHTING?

- WELL, YOU CAN'T SEE THE FLOWER.

IT'S NOT BRIGHT ENOUGH.

- IT'S CALLED "UNDEREXPOSED," AND OF COURSE, OUR GOAL IS TO GET THE CORRECT LIGHTING.

- A LOT OF VARIABLES AFFECT THE WAY A SHOT TURNS OUT, BUT TWO IMPORTANT ONES ARE INTENSITY AND DISTANCE FROM THE SOURCE TO THE SUBJECT.

SO YOUR CHALLENGE TODAY IS TO FIGURE OUT THE RELATIONSHIP BETWEEN DISTANCE AND LIGHT INTENSITY.

KNOWING THAT CAN HELP YOU FIGURE OUT HOW FAR TO PLACE THE LIGHT SOURCE FROM THE FLOWER TO GET THE RIGHT EXPOSURE.

- EACH TEAM IS GONNA GET A LIGHT, A LIGHT PROBE, AND A TAPE MEASURE, AND YOU'LL WANT TO RECORD LIGHT READINGS AT DIFFERENT DISTANCES FROM THE LIGHT, AND THEN USE YOUR DATA TO FIND THE MATHEMATICAL RELATIONSHIP BETWEEN THE TWO VARIABLES, LIGHT INTENSITY AND DISTANCE.

GOOD LUCK.

- DO YOU WANT TO DO IT EVERY 30 INCHES?

- SURE.

ALL RIGHT.

- ALL RIGHT, SO WE'RE ON 30 INCHES.

- THE LIGHT LEVEL IS 1.006.

0.2411.

0.1169.

0.0602.

0.0399.

- OKAY, AWESOME.

- I'M GUESSING WE SHOULD TRY MAYBE EVERY 2 FEET.

- OKAY.

- AND RIGHT NOW, AT 2 FEET, IT READS 1.0006.

0.48.

0.17.

0.098.

0.06.

0.045.

- SO THAT'S THE LAST ONE.

- OKAY.

I SAY THAT WE START OFF BY MAYBE GRAPHING IT USING DISTANCE AS OUR CHANGE IN "X" AND LIGHT INTENSITY AS OUR CHANGE IN "Y."

- OKAY.

- OKAY.

SO WE CAN SEE THAT AS THE DISTANCE IS INCREASING, THE LIGHT INTENSITY IS DECREASING, SO THERE'S AN INVERSE RELATION.

- OKAY.

SO WE CAN COME UP WITH AN EQUATION FROM THERE.

SO WE CAN SAY THAT THE LIGHT INTENSITY... - UH-HUH.

"I" TIMES "D," WHICH IS THE DISTANCE, IS EQUAL TO "K." - AND "K" IS THE CONSTANT.

- RIGHT.

I THINK FROM THERE, JUST TO MAKE IT A LITTLE EASIER, WE CAN MAYBE TURN IT INTO "I" OVER-- NO, I'M SORRY, "I" IS EQUAL TO "K" OVER "D." - WHEN WE PLOTTED THE VALUES, WE NOTICED A CURVE AND NOT A LINE.

IT'S DECREASING MORE QUICKLY IN THE BEGINNING AND THEN-- - AND THEN SLOWING DOWN TOWARDS THE END, RIGHT.

IN ORDER TO MAKE "I" GO DOWN LIKE IT IS IN OUR GRAPH, WE HAVE TO CHANGE THE DISTANCE, SO... HMM.

- OH, I SEE WHAT YOU'RE SAYING.

SO BECAUSE THE DENOMINATOR IS DISTANCE IN THIS CASE, WE WANT IT TO GET BIGGER QUICKLY SO THAT THE ENTIRE VALUE OF "I" DECREASES QUICKLY AS WELL.

- RIGHT, EXACTLY.

SO WE NEED TO RAISE IT TO A POWER.

- WANT TO TRY THE POWER OF TWO?

- OKAY, SURE, LET'S GIVE IT A TRY.

- OKAY, SO WE'RE STATING THAT "I" EQUALS "K" OVER "D" SQUARED?

- MM-HMM.

- LET'S TAKE A LOOK AT THE DISTANCES 4 FEET AND 8 FEET AND SEE WHAT LIGHT INTENSITIES WE GOT FOR THOSE AND HOW THEY WORK WITH THIS RELATIONSHIP.

- MM-HMM.

- SO IF WE DO 0.48 TIMES 16, WE GET 7.68, AND IF WE 0.098 TIMES 64, WE GET ABOUT 6.3.

- ONE THING WE HAD TO REMEMBER WHEN DOING THIS PROBLEM WAS THAT WE HAD TO ALLOW A MARGIN OF ERROR BECAUSE WE WERE USING REAL-LIFE DATA.

- AND BECAUSE OF THE MARGIN OF ERROR, THEY'RE BOTH ABOUT-- WE COULD SAY 7.

- OKAY.

- WHICH MEANS THAT OUR RELATIONSHIP SOUNDS ABOUT RIGHT, BUT WE CAN VERIFY THAT WITH DIFFERENT NUMBERS.

- I WAS THINKING MAYBE WE COULD TRY FOR 12 ALSO.

- OKAY, 0.045 TIMES 144 EQUALS ABOUT 7, WHICH MEANS THAT OUR RELATIONSHIP SOUNDS ABOUT RIGHT.

- OUR FIRST MEASUREMENT OF LIGHT INTENSITY IS OVER THE VALUE OF 1.

WE CAN CROSS THAT OUT BECAUSE IT'S INACCURATE, OKAY?

SO NOW WE HAVE FOUR POINTS TO WORK WITH.

WE ALSO CAN SEE THAT AS THE DISTANCE INCREASES, THE LIGHT INTENSITY SEEMS TO DECREASE, SO IT WOULD BE INVERSE RELATIONSHIP, RIGHT?

- YES.

- DO YOU WANT TO START GRAPHING IN ORDER TO SEE WHAT OTHER KIND OF RELATIONSHIPS WE CAN FIND?

- SURE.

THE LIGHT INTENSITY IS "Y."

AND THE DISTANCE IS "X."

- OKAY.

SO WE ALREADY SAID THAT THE RELATIONSHIP BETWEEN THE DISTANCE AND THE LIGHT INTENSITY IS AN INVERSE RELATIONSHIP.

- YEAH.

- AND IF WE LOOK AT OUR GRAPH, IT LOOKS LIKE IT'S AN EXPONENTIAL INVERSE RELATIONSHIP.

- SO AS THE DISTANCE INCREASES, THE LIGHT INTENSITY DECREASES AT A FASTER RATE.

- RIGHT.

- SO IF WE KNOW THAT IT'S AN EXPONENTIAL RELATIONSHIP, THEN IF WE TOOK 60 INCHES AND 120 INCHES, WHICH IS DOUBLE, THEN WE ALSO SEE THAT BETWEEN THE LIGHT INTENSITY OF 0.24 AND 0.06, THAT'S 1/4.

- OKAY, SO AS WE SEE WITH THIS RELATIONSHIP BETWEEN 60 INCHES AND 120 INCHES, WE DOUBLED THE DISTANCE, BUT THE LIGHT INTENSITY DECREASED BY 1/4.

AND AS WE SAW WITH OUR GRAPH, IT'S AN EXPONENTIAL RELATIONSHIP AS WELL AS AN INVERSE RELATIONSHIP.

SO IF WE PUT THIS ALL TOGETHER, WE GET THAT THE LIGHT INTENSITY IS EQUAL TO SOME CONSTANT OVER THE DISTANCE SQUARED.

- SO WHAT DID YOU GUYS COME UP WITH?

- SO AS WE MOVED THE LIGHT AWAY FARTHER FROM THE FLOWER AND INCREASED THE DISTANCE, WE NOTICED THAT THE LIGHT INTENSITY DECREASED AT A FASTER RATE.

- AND BASED ON THE INFORMATION, WE SAW THAT THE INTENSITY OF THE LIGHT WAS EQUAL TO A CONSTANT, "K," DIVIDED BY THE DISTANCE SQUARED.

- SO IT'S AN INVERSE SQUARE RELATIONSHIP.

- NICE JOB SOLVING THE CHALLENGE.

BASED ON YOUR ANSWER, WHAT DO YOU THINK YOU COULD'VE DONE TO HAVE FIXED THAT UNDEREXPOSED SHOT?

LET'S ASSUME THAT WE WANT THE LIGHT INTENSITY TO BE 4 TIMES WHAT IT WAS IN THAT TAKE.

- WELL, THEN YOU WOULD DECREASE THE DISTANCE BY 1/2.

- EXACTLY, DO YOU WANT TO SEE THE FINAL SHOT?

all: YEAH.

- SURE.

[pulsing rock music] - THAT'S AWESOME.

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