- A CERTAIN RADIOACTIVE MATERIAL DECAYS IN SUCH A WAY
THAT THE MASS IN KILOGRAMS REMAINING AFTER 2 YEARS
IS GIVEN BY THE FUNCTION M OF T EQUALS 120 TIMES E RAISED
TO THE POWER OF NEGATIVE 0.018 TIMES T.
THE GENERAL FORM FOR AN EXPONENTIAL DECAY MODEL IS P
OF T EQUALS P SUBZERO OR P NOT,
TIMES E RAISED TO THE POWER OF NEGATIVE KT,
WHERE T IS THE TIME,
K IS THE DECAY RATE EXPRESSED AS A DECIMAL,
AND P SUBZERO OR P NOT IS THE INITIAL AMOUNT.
THIS IS ALMOST THE SAME AS EXPONENTIAL GROWTH EXCEPT
FOR EXPONENTIAL DECAY,
THE EXPONENT IS NEGATIVE HERE
AND THEREFORE, THE FUNCTION WOULD BE DECREASING
SO, GOING BACK TO OUR EXAMPLE, OUR FIRST QUESTION IS,
WHAT IS THE DECAY RATE EXPRESSED AS A PERCENTAGE.
AGAIN, THE DECAY RATE IS K,
SO HERE THE DECAY RATE WOULD BE 0.018 EXPRESSED AS A DECIMAL.
SO, WE HAVE K EQUALS 0.018, BUT WE'RE ASKED TO EXPRESS THIS
AS A PERCENTAGE, SO WE MULTIPLY BY 100 AND ADD A PERCENT SIGN,
OR JUST MOVE THE DECIMAL POINT TO THE RIGHT TWO PLACES
AND ADD A PERCENT SIGN.
THIS WOULD BE 1.8 PERCENT.
SO, WE CAN SAY THE DECAY RATE IS 1.8 PERCENT PER YEAR.
WE DON'T INCLUDE THE NEGATIVE SIGN HERE
BECAUSE I'VE ALREADY STATED THAT IT IS THE K.
NEXT, WE WANT TO KNOW WHAT THE INITIAL MASS IS
WHEN TIME T EQUALS ZERO.
WE SHOULD RECOGNIZE THIS IS GOING TO BE P SUBZERO
OR IN THIS CASE, 120 KILOGRAMS,
BUT JUST TO SHOW WHY, LET'S GO AHEAD AND EVALUATE THIS FUNCTION
AT T EQUALS ZERO.
SO, M OF ZERO WOULD BE EQUAL
TO 120 TIMES E RAISED TO THE POWER OF NEGATIVE 0.018
TIMES ZERO.
SO, NOTICE HERE, THE EXPONENT ON E IS ZERO.
SO, WE HAVE 120 TIMES E TO THE ZERO.
ANYTHING TO THE ZERO POWER IS EQUAL TO 1, SO THIS IS JUST 120.
SO, THE INITIAL MASS IS 120 KILOGRAMS.
AND NOW FOR THE LAST QUESTION,
HOW MUCH MASS REMAINS AFTER 50 YEARS,
ROUNDED TO TWO DECIMAL PLACES?
WELL, THEY'RE TELLING US THAT TIME T EQUALS 50 YEARS,
SO WE SUBSTITUTE 50 FOR T OR FIND THE VALUE OF M OF 50,
WHICH WOULD BE 120 TIMES E RAISED TO THE POWER
OF NEGATIVE 0.018 TIMES 50.
AND NOW, WE'LL GO TO THE CALCULATOR
TO GET A DECIMAL APPROXIMATION.
WE HAVE 120.
NOW HERE, IF WE PRESS SECOND LN,
IT BRINGS UP E RAISED TO THE POWER OF
AND THEN WE HAVE NEGATIVE 0.018 TIMES 50,
CLOSE PARENTHESIS AND ENTER.
IF WE ROUND TO TWO DECIMAL PLACES,
NOTICE HOW THE THIRD DECIMAL PLACE IS AN 8,
WHICH MEANS, ROUND UP.
THIS WOULD BE APPROXIMATELY 48.79 KILOGRAMS.
SO, AFTER 50 YEARS,
APPROXIMATELY 48.79 KILOGRAMS REMAIN.
BEFORE WE GO, LET'S TAKE A LOOK AT THE GRAPH OF OUR FUNCTION.
NOTICE HOW FOR EXPONENTIAL DECAY,
THE FUNCTION IS DECREASING.
WHEN WE FOUND THE INITIAL AMOUNT OR WHEN T EQUALED ZERO,
WE FOUND THE FUNCTION VALUE HERE AT 120
AND THEN WHEN WE FOUND THE AMOUNT REMAINING AFTER 50 YEARS,
WE FOUND THE FUNCTION VALUE HERE AT T EQUALS 50.
I HOPE YOU FOUND THIS HELPFUL.