- A HOUSE PURCHASE FOR 110,000 IN 1995
IS VALUED AT 162,000 IN 2008.
WE WANT TO ASSUME EXPONENTIAL GROWTH
AND THEN FIND THE ANNUAL GROWTH RATE FROM 1995 TO 2008.
AND THEN, IF THE VALUE CONTINUES TO GROW
BY THE SAME RATE,
WHAT WILL THE HOUSE BE WORTH IN THE YEAR 2012.
SO TO MODEL THE VALUE OF THE HOUSE,
WE'LL USE THE EXPONENTIAL FUNCTION F(X) = A x B
RAISED TO THE POWER OF X,
WHERE B, THE BASE, IS EQUAL TO 1 + R,
WHERE R IS EQUAL TO THE EXPONENTIAL GROWTH
OR DECAY RATE, EXPRESSED AS A DECIMAL.
"A" IS THE INITIAL AMOUNT,
OR THE FUNCTION VALUE WHEN X EQUALS ZERO
AND X IS THE EXPONENT,
OR IN THIS CASE, X WILL BE THE TIME IN YEARS.
SO GOING BACK UP TO OUR GIVEN INFORMATION,
SINCE THE HOUSE WAS PURCHASED FOR 110,000 IN 1995,
WE'RE GOING TO USE 1995 AS THE BASE YEAR
WHICH MEANS WE'LL LET T = 0 REPRESENT 1995.
AND THIS IS A COMMON PRACTICE
WHEN THE TIME IS GIVEN AS A YEAR.
SO IF THE HOUSE IS VALUED AT 110,000 IN 1995,
WE CAN NOW SAY THAT F(0) MUST BE EQUAL TO 110,000.
THIS IS HELPFUL BECAUSE NOW WE KNOW
THAT "A", THE INITIAL VALUE,
IS GOING TO BE EQUAL TO 110,000.
NEXT, THE VALUE OF THE HOUSE IS 162,000 IN THE YEAR 2008,
AND SINCE 2008 MINUS 1995 IS EQUAL TO 13,
THIS IS TELLING US THAT F(13) IS EQUAL TO 162,000.
SO AGAIN, T REPRESENTS THE NUMBER OF YEARS AFTER 1995.
AND NOW FOR OUR FUNCTION,
SINCE WE'RE USING THE VARIABLE T, FOR TIME,
F(T) MUST BE EQUAL TO THE INITIAL VALUE
OR 110,000 x B RAISED TO THE POWER OF T.
NOTICE THAT WHEN T IS EQUAL TO ZERO,
THE FUNCTION VALUE WOULD BE 110,000
BECAUSE B TO THE ZERO IS EQUAL TO ONE.
AND NOW WE CAN USE THE SECOND FUNCTION VALUE
TO DETERMINE THE VALUE OF B.
IF F(13) IS EQUAL TO 162,000,
THEN F(13) WOULD BE EQUAL TO 110,000 x B TO THE 13th,
WHICH MUST EQUAL 162,000.
AND NOW WE'LL SOLVE THIS FOR B TO DETERMINE OUR BASE.
SO WE'LL DIVIDE BOTH SIDES BY 110,000.
SO WE HAVE B TO THE 13th IS EQUAL TO THIS FRACTION HERE.
AND TO SOLVE THIS FOR B, WE'RE GOING TO RAISE BOTH SIDES
TO THE RECIPROCAL POWER OF 13,
WHICH WOULD BE 1/13th.
REMEMBER WHEN RAISING POWERS TO POWERS,
WE MULTIPLY THE EXPONENTS.
13 x 1/13th = B TO THE 1st, OR JUST B.
AND NOW WE'LL FIND THIS VALUE HERE ON THE CALCULATOR.
WE DO NEED THIS FRACTION IN PARENTHESES.
WE'RE GOING TO RAISE THIS TO THE 1/13th POWER.
AGAIN, THE 1/13th WILL BE IN PARENTHESES.
ROUNDING TO FOUR DECIMAL PLACES, THIS WILL BE 1.0302
WHICH MEANS THE EXPONENTIAL FUNCTION THAT WE'LL USE
TO DETERMINE THE VALUE OF THE HOUSE,
WHEN T IS THE NUMBER OF YEARS AFTER 1995,
WILL BE F(T) = 110,000 x 1.0302
RAISED TO THE POWER OF T.
NOW TO ANSWER THIS FIRST QUESTION,
WHAT IS THE ANNUAL GROWTH RATE FROM 1995 TO 2008,
REMEMBER THE BASE IS EQUAL TO 1 + R,
WHERE R IS THE GROWTH RATE EXPRESSED AS A DECIMAL.
SO IN THIS CASE, SO IN THIS CASE, 1.0302 MUST EQUAL 1 + R.
SO SUBTRACTING ONE ON BOTH SIDES OF THE EQUATION,
WE HAVE R IS EQUAL TO 0.0302.
EXPRESSED AS A DECIMAL,
OUR GROWTH RATE IS 3.02% PER YEAR FROM 1995 TO 2008.
AND NOW FOR THE LAST QUESTION,
IF THE VALUE CONTINUES TO GROW BY THE SAME RATE,
WHAT WILL THE HOUSE BE WORTH IN THE YEAR 2012.
THE FIRST STEP HERE IS TO DETERMINE THE VALUE OF T
FOR THE YEAR 2012.
WELL, T IS GOING TO BE EQUAL TO THE DESIRED YEAR OF 2012
MINUS THE BASE YEAR OF 1995.
SO THIS IS GOING TO BE EQUAL TO 17.
SO TO ANSWER THIS QUESTION, WE NEED TO FIND F(17),
WHICH IS EQUAL TO 110,000 x 1.0302
RAISED TO THE 17th POWER.
WE'LL GO BACK TO THE CALCULATOR.
RAISE THIS TO THE 17th POWER.
WE'LL GO AHEAD AND ROUND TO THE NEAREST DOLLAR
AND SAY THE VALUE IS GOING TO BE APPROXIMATELY $182,414.
BUT OF COURSE, AS WE ALL KNOW BY NOW,
HOME PRICES DON'T ALWAYS INCREASE OVER TIME.
SOMETIMES THEY CAN GO DOWN IN VALUE
AND THAT'S WHY EVERY MODEL DOES HAVE IT'S LIMITATIONS.
BUT BASED UPON THE INFORMATION FROM THIS PROBLEM,
IF THE TREND CONTINUES,
THIS WOULD BE THE APPROXIMATE VALUE OF THE HOUSE
IN THE YEAR 2012.
I HOPE YOU FOUND THIS HELPFUL.