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- A BASEBALL TEAM PLAYS IN A STADIUM
THAT HOLDS 66,000 SPECTATORS.
WITH A TICKET PRICE AT $9.00 THE AVERAGE ATTENDANCE
HAS BEEN 28,000; WHEN THE PRICE DROPPED TO $6.00
THE AVERAGE ATTENDANCE ROSE TO 33,000.
FIND A DEMAND FUNCTION, D OF Q, WHERE Q IS THE QUANTITY
OR NUMBER OF SPECTATORS AND WE CAN ASSUME
THAT D OF Q IS LINEAR.
SO BECAUSE D OF Q IS LINEAR WE'LL FIND
THE EQUATION OF D OF Q IN SLOPE INTERCEPT FORM
WITH THE FORM Y = MX + B, BUT BECAUSE OUR FUNCTION
IS D OF Q WE WOULD HAVE D OF Q = M x Q + B.
SO FROM THE GIVEN INFORMATION WE'LL FIND THE SLOPE OF THE LINE
AND THEN FROM THERE WE CAN USE POINT SLOPE FORM OF THE LINE
TO FIND D OF Q, BUT BECAUSE OUR FUNCTION IS D OF Q
EACH ORDERED PAIR WOULD BE Q, D OF Q.
WELL IF WE WANT Q, P FOR PRICE.
SO THE TICKET PRICE OF $9.00 AND THE ATTENDANCE AT 28,000
THE ORDERED PAIR WOULD NOT BE 9, 28,000; IT WOULD BE 28,000, 9.
AND WHEN THE PRICE IS $6.00 THE ATTENDANCE IS 33,000;
WHICH MEANS THE ORDERED PAIR WOULD BE 33,000, 6.
IT'S IMPORTANT THAT WE HAVE THESE IN THE CORRECT ORDER
WHERE THE FIRST COORDINATE IS THE QUANTITY
OR NUMBER OF SPECTATORS AND SECOND COORDINATE
WOULD BE D OF Q OR THE PRICE P.
NOW THAT WE HAVE THESE 2 ORDERED PAIRS WE CAN FIND THE SLOPE
WHERE THE SLOPE IS = TO THE CHANGE OF Y
DIVIDED BY THE CHANGE OF X.
SO WE CAN CALL THIS X SUB 1, Y SUB 1, X SUB 2, Y SUB 2.
AND THEREFORE THE SLOPE WOULD BE = TO THE CHANGE OF Y,
WHICH WOULD BE 6 - 9 DIVIDED BY THE CHANGE OF X.
SO THIS WOULD BE 33,000 - 28,000 AND THEREFORE
THE SLOPE OF THE LINE IS = TO -3 DIVIDED BY 5,000.
SO NOW THAT WE HAVE THE SLOPE WE CAN USE THIS
AND ONE OF THESE 2 POINTS TO FIND THE EQUATION
IN POINT SLOPE FORM AND THEN SOLVE FOR Y,
IN OUR CASE, SOLVE FOR D OF Q.
SO IF USE THIS SLOPE AND THIS POINT,
THE EQUATION OF OUR FUNCTION WOULD BE INSTEAD OF Y
WE WOULD HAVE D OF Q - Y SUB 1 OR - 9 = THE SLOPE,
WHICH IS -3 DIVIDED BY 5,000 x THE QUANTITY X - X SUB 1
OR IN OUR CASE Q - 28,000.
NOW LETS SOLVE THIS FOR D OF Q ON THE NEXT SLIDE.
LET'S BEGIN BY ADDING 9 TO BOTH SIDES.
THAT WOULD GIVE US D OF Q = -3
DIVIDED BY 5,000 x THE QUANTITY Q - 28,000 + 9.
NOW FOR OUR HOMEWORK WE CAN LEAVE IT IN THIS FORM HERE,
BUT I'M ALSO GOING TO SHOW HOW WE CAN DISTRIBUTE
AND COMBINE LIKE TERMS.
AGAIN, THIS WOULD BE ONE OPTION FOR OUR FUNCTION TO GIVE Q.
IF WE DID CONTINUE THE NEXT STEP WOULD BE TO DISTRIBUTE
THE FRACTION HERE.
SO WE'D HAVE D OF Q = -3 DIVIDED BY 5,000 x Q.
NEXT, WE HAVE THIS NEGATIVE FRACTION x -28,000,
WHICH WE'LL USE A CALCULATOR FOR.
-3 DIVIDED BY 5,000 x -28,000; IT'S +16.8.
LET'S CONVERT THIS TO A FRACTION THOUGH SINCE OUR SLOPE
IS A FRACTION.
SO MATH ENTER, ENTER.
SO WE HAVE +84/5 OR + 84/5 + 9/1.
OUR COMMON DENOMINATOR HERE IS 5 AND MULTIPLY 9/1 BY 5/5.
SO WE HAVE D OF Q = -3 DIVIDED BY 5,000 x Q
AND THIS WILL BE + 84/5 + 45/5, WHICH WOULD BE 129/5.
SO AGAIN WE CAN GO AHEAD AND LEAVE THE DEMAND FUNCTION
IN THIS FORM HERE OR IF WE WANT WE CAN WRITE IT
IN A SLOPE INTERCEPT FORM, WHICH WOULD BE THIS FORM HERE.
I HOPE YOU FOUND THIS HELPFUL.