- A RECTANGLE HAS AN AREA OF 176 SQUARE CENTIMETERS
AND A PERIMETER OF 54 CENTIMETERS
OR THE DIMENSIONS OF THE RECTANGLE.
USING THIS RECTANGLE HERE LET'S LABEL THIS LENGTH X CENTIMETERS.
LET'S LABEL THE LENGTH OF THIS SIDE Y CENTIMETERS.
AND BECAUSE THE AREA FORMULA OF A RECTANGLE
IS LENGTH TIMES WIDTH
WE KNOW THAT X x Y = 176.
WE ALSO KNOW THE PERIMETER IS EQUAL TO 54 CENTIMETERS
WHERE THE PERIMETER WOULD THE SUM OF THE LENGTHS
OF THE FOUR SIDES
AND, THEREFORE, 2X + 2Y = 54.
AND NOW IF WE SOLVE THIS SYSTEM OF EQUATIONS FOR X AND Y
WE CAN DETERMINE THE DIMENSIONS OF THE RECTANGLE
THAT HAVE THE GIVEN AREA AND THE GIVEN PERIMETER.
LET'S SOLVE THIS SYSTEM OF EQUATIONS USING SUBSTITUTION.
LET'S BEGIN BY SOLVING EQUATION 2 FOR Y.
TO DO THIS THE FIRST STEP WOULD BE TO SUBTRACT 2X ON BOTH SIDES
GIVING US THE EQUATION 2Y = 54 - 2X.
AND NOW WE'LL DIVIDE BOTH SIDES BY 2.
SO WE KNOW THAT Y MUST EQUAL 27 - X.
AND SINCE Y IS EQUAL TO 27 - X WE CAN REPLACE Y WITH 27 - X
IN THIS FIRST EQUATION
AND THEN SOLVE FOR X.
SO BY SUBSTITUTION WE WOULD HAVE THE EQUATION X(27 - X) = 176.
NO WE'LL CLEAR PARENTHESES BY DISTRIBUTING X
THAT WOULD GIVE US 27X - X SQUARED = 176.
NOTICE WE HAVE A QUADRATIC EQUATION
SO WE'LL SET THIS EQUAL TO ZERO AND SOLVE FOR X.
WE PREFER TO HAVE THE X SQUARED TERM POSITIVE
SO LET'S GO AHEAD AND ADD X SQUARED TO BOTH SIDES,
AS WELL AS SUBTRACT 27X ON BOTH SIDES.
THAT WOULD GIVE US THE EQUATION 0 = X SQUARED - 27X + 176.
NOW, IF THIS DOES FACTOR IT'LL FACTOR INTO TWO BINOMIAL FACTORS
WHERE THE FIRST POSITIONS MUST COME FROM THE FACTORS
OF X SQUARED
WHICH ARE X AND X.
THE SECOND TERMS WOULD BE THE FACTORS OF 176 THAT ADD THE -27.
THAT'S NOT AN EASY QUESTION BUT IF WE START TO BREAK DOWN 176
INTO ITS PRIME FACTORS
WE CAN START WITH 2 x 88, CAN BREAK DOWN 88 INTO 8 x 11.
AND NOTICE IF WE STOP HERE THIS SHOWS US THAT 16 X 11 = 176
AND 16 + 11 = 27.
AND SINCE WE WANT -27 WE COULD USE A FACTOR OF -16
AND A FACTOR OF -11 WHICH WOULD GIVE US A PRODUCT OF 176
AND WOULD SUM TO -27.
SO NOTICE HOW WE HAVE TWO SOLUTIONS TO THIS EQUATION.
WE HAVE X = +16 OR X = +11.
IT MAY SEEM STRANGE TO HAVE TWO POSSIBLE VALUES FOR X
BUT NOTICE HOW IF X IS 16,
Y = 27 - 16 OR 11.
OR IF X IS 11 THEN Y = 27 - 11 OR 16.
EITHER WAY WE'LL HAVE THE SAME DIMENSIONS FOR THE RECTANGLE.
BUT FROM OUR DIAGRAM LOOKS LIKE X IS LONGER
SO LET'S LET THIS LENGTH BE 16 CENTIMETERS
AND, THEREFORE, THIS LENGTH HERE WHICH IS Y CENTIMETERS
WOULD BE 27 - 16 OR 11 CENTIMETERS.
AND, THEREFORE, WE CAN SAY THE DIMENSIONS OF THE RECTANGLE
WOULD BE 16 CENTIMETERS BY 11 CENTIMETERS.
A RECTANGLE WITH THESE DIMENSIONS WOULD HAVE AN AREA
OF 176 SQUARE CENTIMETERS
AND A PERIMETER OF 54 CENTIMETERS.
I HOPE YOU FOUND THIS HELPFUL.