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- AN AIRPLANE IS FLYING FROM SAN FRANCISCO, CALIFORNIA
TO DENVER, COLORADO.
AS IT BEGINS TO DESCEND,
THE ALTITUDE OF THE PLANE IS GIVEN BY THE EQUATION
A = 33,000 - 120T,
WHERE A IS THE ALTITUDE IN FEET,
AND T IS THE TIME IN SECONDS.
WE WANT TO ANSWER THE FOLLOWING QUESTIONS.
NUMBER ONE, WHAT IS THE SLOPE OF THE LINE
AND WHAT DOES IT REPRESENT.
REMEMBER IF A LINE IS WRITTEN IN SLOPE-INTERCEPT FORM
OR THE FORM Y = MX + B,
THE COEFFICIENT OF X WOULD BE THE SLOPE.
SO IF WE WANTED OUR LINEAR EQUATION
TO FIT THIS FORM EXACTLY,
WE COULD WRITE THIS AS A =-- WE'LL PUT THE X TERM FIST
SO IT'LL BE -120T AND THEN + 33,000.
SO IN THIS FORM WE SHOULD BE ABLE TO RECOGNIZE
THAT THE SLOPE IS EQUAL TO -120.
IF WE WANT TO MAKE MEANING OF THIS SLOPE,
IT'S HELPFUL TO PUT IT OVER 1.
AND NORMALLY A SLOPE IS THE RATIO OF THE CHANGE OF Y
TO THE CHANGE OF X,
BUT IN THIS CASE THE CHANGE OF Y
IS ACTUALLY THE CHANGE IN THE ALTITUDE,
AND THE CHANGE OF X IS ACTUALLY THE CHANGE OF T.
SO THE SLOPE TELLS US THE PLANE'S ALTITUDE IS DECREASING
AT 120 FEET PER 1 SECOND.
AND WE KNOW IT'S DESCENDING
BECAUSE THE CHANGE IN THE ALTITUDE IS NEGATIVE.
SO LET'S GO AHEAD AND WRITE THIS OUT.
SLOPE REPRESENTS THE RATE AT WHICH THE PLANE IS DESCENDING,
WHICH IS 120 FEET PER SECOND.
NUMBER TWO, WHAT IS THE Y INTERCEPT
AND WHAT DOES IT REPRESENT.
WELL, THE Y INTERCEPT IS THE VALUE OF B,
SO THE Y INTERCEPT = 33,000.
REMEMBER THE Y INTERCEPT OCCURS WHEN X = 0,
OR IN THIS CASE, WHEN T = 0.
WE'LL LET T = 0,
THAT'S AT THE INSTANT THE PLANE STARTS TO DESCEND.
SO T = THE HEIGHT OF THE PLANE
RIGHT BEFORE IT BEGINS TO DESCEND.
SO THE Y INTERCEPT IS 33,000, AND THIS SHOULD BE IN FEET,
WHICH REPRESENTS THE ALTITUDE OF THE PLANE
RIGHT BEFORE IT DESCENDS.
NUMBER THREE ASKS, WHAT IS THE ALTITUDE AFTER THREE MINUTES.
WELL, AGAIN, WE'RE GIVEN THE EQUATION
THAT TELLS US THE ALTITUDE WHEN T IS TIME IN SECONDS.
SO THE ONE THING WE HAVE TO BE CAREFUL ABOUT HERE
IS THAT THE TIME IS GIVEN IN MINUTES,
BUT THE EQUATION REQUIRES TIME IN SECONDS.
SO WE FIRST NEED TO CONVERT 3 MINUTES TO SECONDS.
AND SINCE THERE ARE 60 SECONDS IN 1 MINUTE,
WE CAN SEE HERE THAT WE'RE GOING TO USE T = 180 SECONDS.
SO WE'LL HAVE THE ALTITUDE IS GOING TO BE EQUAL TO--
WE'LL GO AHEAD AND USE THE ORIGINAL EQUATION,
33,000 - 120 x 180.
SO WE'LL HAVE 33,000 - THIS PRODUCT, WHICH IS 21,600.
AND THIS DIFFERENCE IS 11,400, AND THIS WOULD BE IN FEET.
SO AFTER THREE MINUTES THE ALTITUDE OF THE PLANE
IS 11,400 FEET.
AND THEN FOR NUMBER FOUR, ONCE THE PLANE STARTS TO DESCEND
HOW LONG WILL IT TAKE TO LAND?
WELL, GOING BACK TO OUR EQUATION,
A REPRESENTS THE ALTITUDE OF THE PLANE.
SO RIGHT WHEN THE PLANE LANDS THE ALTITUDE WOULD BE ZERO.
SO IF WE SET A = 0 AND SOLVE FOR T
WE CAN ANSWER NUMBER FOUR.
SO WE'LL ADD 120T TO BOTH SIDES.
SO WE HAVE 120T = 33,000.
LET'S GO AHEAD AND FINISH IT OVER HERE.
SO WE'LL DIVIDE BOTH SIDES BY 120,
SO WE'LL HAVE T = THIS QUOTIENT,
AND 33,000 DIVIDED BY 120 = 275.
AND THIS WOULD BE SECONDS.
IF WE WANT TO CONVERT THIS TO MINUTES,
SINCE THERE ARE 60 SECONDS IN EVERY MINUTE,
AND 60 x 4 IS 240,
THIS WOULD BE = 4 MINUTES AND 35 SECONDS.
AGAIN, 6 x 4 WOULD BE 240 SECONDS
LEAVING US WITH AN EXTRA 35 SECONDS.
I HOPE YOU FOUND THIS HELPFUL.