- HERE'S AN APPLICATION PROBLEM
INVOLVING REVENUE COSTS AND PROFIT
THAT CAN BE SOLVED USING A LINEAR EQUATION.
A COMPANY MAKES PHONE COVERS.
THEY SELL EACH COVER FOR $12.
IF X EQUALS THE NUMBER OF PHONE COVERS
PRODUCED AND SOLD,
WHAT IS THE REVENUE EQUATION?
WELL THE REVENUE WOULD BE THE TOTAL AMOUNT OF MONEY
GENERATED FROM THE SALE OF THE PHONE COVERS
SO WE CAN SAY THAT THE REVENUE BIG R,
IS GOING TO BE EQUAL TO 12 x X
WHERE AGAIN X IS THE NUMBER OF PHONE COVERS
PRODUCED AND SOLD.
AND R WOULD BE THE TOTAL REVENUE IN DOLLARS.
NEXT THE COST TO MAKE THE COVERS IS $2 EACH
WITH A ONE TIME START UP COST OF $5,200.
SO NOW WE WANT TO KNOW WHAT THE COST EQUATION IS.
AND WE CAN SEE THERE ARE TWO COMPONENTS TO THE COST,
THERE'S THIS FIXED COST OF $5,200
THAT'S NOT GOING TO CHANGE
AND THEN WE HAVE A VARIABLE COST
BASED UPON THE NUMBER OF PHONE COVERS PRODUCED AT $2 EACH.
SO THE TOTAL COST IS GOING
TO BE EQUAL TO 2 x THE NUMBER OF PHONE COVERS PRODUCED,
PLUS THE FIXED COSTS, OR START UP COSTS OF $5,200.
SO C IS GOING TO REPRESENT THE TOTAL COST IN DOLLARS
WHERE X IS THE NUMBER OF PHONE COVERS PRODUCED.
AND IT'S PRETTY TYPICAL TO LEAVE THE UNITS OFF
IN THE EQUATION
AND JUST MAKE A NOTE FOR WHAT THE VARIABLES REPRESENT.
NOW WE WANT TO DETERMINE THE PROFIT
WHEN THE COMPANY SELLS 1,200 COVERS.
SO THE FIRST THING WE SHOULD DO
IS WRITE THE PROFIT EQUATION
AND PROFIT IS EQUAL TO REVENUE MINUS COSTS.
AGAIN WHERE REVENUE IS THE AMOUNT OF MONEY COMING IN
AND THE TOTAL COST IS THE AMOUNT OF MONEY GOING OUT.
SO THE PROFIT, BIG P,
IS GOING TO BE EQUAL TO THE REVENUE EQUATION
OF 12X - THE COST EQUATION OF 2X + 5,200.
NOW IT IS IMPORTANT THAT WE ADD PARENTHESIS HERE
BECAUSE WE DO HAVE TO SUBTRACT THE ENTIRE COST EQUATION.
WE DO HAVE LIKE TERMS HERE AND HERE
SO WE CAN SIMPLIFY THIS EQUATION.
WE'RE GOING TO HAVE THE PROFIT,
IT'S GOING TO BE EQUAL TO 12X - 2X, THAT WOULD BE 10X
AND THEN THIS WOULD BE MINUS 5,200.
SO NOW WE CAN USE THIS EQUATION
TO DETERMINE THE PROFIT
WHEN THE COMPANY SELLS 1,200 COVERS.
THE PROFIT WOULD BE EQUAL TO 10 x 1,200 - 5,200.
WELL HERE WE'D HAVE 12,000 - 5,200
SO THIS DIFFERENCE WOULD BE THE COMPANY'S PROFIT
FROM SELLING 1,200 COVERS
SO THE PROFIT IS GOING TO BE EQUAL TO--
THIS WOULD BE $6,800.
AND THE LAST QUESTION ASKS US TO FIND THE NUMBER OF COVERS
THEY NEED TO SELL TO BREAK EVEN.
WELL BREAK EVEN IS WHEN THE COMPANY DOESN'T MAKE ANY MONEY
OR LOSE ANY MONEY.
SO THAT WOULD OCCUR
WHEN THE REVENUE IS EXACTLY EQUAL TO THE COST.
SO TO DETERMINE THE BREAK EVEN POINT
WE'LL HAVE TO SET THE REVENUE EQUAL TO THE COST
WHICH MEANS WE'LL HAVE 12X = TO 2X + 5,200.
SO TO SOLVE FOR X HERE, WE'LL SUBTRACT 2X ON BOTH SIDES.
WE WOULD HAVE 10X = 5,200.
LET'S GO AHEAD AND FINISH THIS UP HERE.
SO WE'LL DIVIDE BOTH SIDES BY 10,
SO WE HAVE X = 5,200 DIVIDED BY 10 WHICH WOULD BE 520.
SO THEY NEED TO SELL 520 COVERS TO BREAK EVEN,
MEANING THEY WON'T MAKE ANY MONEY OR LOSE ANY MONEY.
I HOPE YOU FOUND THIS HELPFUL.