- A TABLET COMPUTER IS WORTH $1,500 NEW.
IT LOSES 12% OF ITS VALUE EACH YEAR.
THE VALUE CAN BE MODELED BY THE FUNCTION V OF T EQUALS A TIMES B
RAISED TO THE POWER OF T,
WHERE V OF T IS THE DOLLAR VALUE
AND T IS THE NUMBER OF YEARS SINCE THE PURCHASE.
WE WANT TO WRITE A POSSIBLE EXPONENTIAL EQUATION
FOR THE VALUE.
SO OUR FUNCTION WILL BE IN THE FORM OF V OF T.
A REPRESENTS THE INITIAL VALUE OF THE COMPUTER
WHICH WILL BE 1,500
AND WE'RE GOING TO HAVE SUM BASE B RAISED TO THE POWER OF T.
SO LET'S TALK ABOUT HOW WE'RE GOING TO FIND THE BASE
OF THIS EXPONENTIAL FUNCTION.
WE KNOW EACH YEAR THE COMPUTER LOSES 12% OF THE VALUE,
WHICH MEANS IT RETAINS 88% OF THE VALUE.
SO AFTER THE FIRST YEAR, THE VALUE WOULD BE 1,500 TIMES 0.88.
THIS 0.88 REPRESENTS THE 88% OF THE VALUE THAT IT RETAINS
AFTER THE FIRST YEAR.
AFTER THE SECOND YEAR
THE COMPUTER LOSES ANOTHER 12% OF THE VALUE OFF THIS AMOUNT
OR RETAINS 88% OF THIS VALUE HERE.
SO AFTER THE SECOND YEAR IT WOULD BE 1,500 TIMES 0.88
TIMES ANOTHER 0.88.
AGAIN, THIS SECOND 0.88 REPRESENTS THE AMOUNT OF VALUE
RETAINED AFTER THE SECOND YEAR
AND THIS PATTERN IS GOING TO CONTINUE.
AFTER THE THIRD YEAR WE'LL HAVE 1,500 TIMES 0.88 TIMES 0.88
TIMES ANOTHER 0.88.
SO THE BASE OF OUR EXPONENTIAL FUNCTION IS GOING TO BE 0.88
WHICH IS THE PERCENT OF VALUE THE COMPUTER RETAINS
AFTER EACH YEAR.
SO TO FIND THE VALUE AFTER THREE YEARS
WE NEED TO DETERMINE VIA 3.
SO WE'LL SUBSTITUTE 3 FOR T
AND EVALUATE THIS ON THE CALCULATOR.
SO WE'LL HAVE 1,500 TIMES 0.88 RAISED TO THE POWER OF 3.
SO THE VALUE OF THE COMPUTER IS APPROXIMATELY 1,022.21
AFTER THREE YEARS.
AND NOW FOR THE LAST QUESTION,
WHEN WILL THE VALUE BE HALF OF ITS ORIGINAL VALUE?
WELL, THE STARTING VALUE IS 1,500
SO HALF OF THIS WOULD BE 750.
SO WE WANT TO SET D OF T TO 750 AND THEN SOLVE FOR T.
SO WE'LL HAVE 750 EQUALS 1,500 TIMES 0.88
RAISED TO THE POWER OF T.
SO WE WANT TO ISOLATE
THE EXPONENTIAL PART OF THIS EQUATION
SO WE'LL DIVIDE BOTH SIDES BY 1,500.
THIS WILL BE ONE 750 DIVIDED BY 1,500 EQUALS 1/2
OR 0.5 EQUALS 0.88 RAISED TO THE POWER OF T.
AND NOW WE CAN USE LOGARITHMS TO SOLVE THIS FOR T.
IF WE TAKE THE NATURAL LOG OF BOTH SIDES OF THE EQUATION,
ON THE RIGHT SIDE WE CAN USE THE POWER PROPERTY OF LOGARITHMS
TO MOVE THIS T TO THE FRONT,
GIVING US NATURAL LOG OF 0.5 EQUALS T TIMES NATURAL LOG 0.88.
NOW, TO SOLVE THIS EQUATION FOR T
WE CAN DIVIDE BOTH SIDES BY NATURAL LOG 0.88.
NOTICE HERE THIS SIMPLIFIES TO A 1 SO THIS QUOTIENT WILL EQUAL T.
SO WE'LL GO BACK TO THE CALCULATOR.
NATURAL LOG 0.5 DIVIDED BY NATURAL LOG 0.88
WILL GIVE US THE VALUE OF T.
WHEN ROUNDED TO THE NEAREST 10TH
THIS WILL BE APPROXIMATELY 5.4 YEARS.
SO AGAIN THE VALUE WILL BE HALF OF THE ORIGINAL VALUE
AFTER 5.4 YEARS
OR APPROXIMATELY 5.4 YEARS.