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- THE DOUBLING PERIOD FOR A BACTERIAL POPULATION
IS 15 MINUTES.
AFTER 75 MINUTES, THE POPULATION IS 72,000.
WE'RE ASKED TO DETERMINE THE INITIAL POPULATION
AND THE POPULATION AFTER FOUR HOURS.
SO BECAUSE WE'RE GIVEN INFORMATION
ABOUT THE DOUBLING TIME WE'RE GOING TO MODEL THE POPULATION
USING AN EXPONENTIAL FUNCTION IN THIS FORM HERE.
P OF T = "A" x B RAISED TO THE POWER OF T
AND BECAUSE WE'RE GIVEN INFORMATION
ABOUT THE DOUBLING TIME WE'RE GOING TO LET THE BASE = 2.
WE COULD ALSO USE THIS FORM OF AN EXPONENTIAL FUNCTION
WITH BASE E BUT FOR THIS EXAMPLE,
WE ARE GOING TO USE THIS FORM HERE.
WHERE P OF T IS THE AMOUNT AFTER TIME T,
"A" IS THE STARTING OR INITIAL AMOUNT,
B IS THE BASE AND T IS THE TIME.
SO LOOKING AT THE GIVEN INFORMATION,
IF THE DOUBLING TIME IS 15 MINUTES
THAT MEANS P OF 15 MUST EQUAL 2 TIMES THE INITIAL AMOUNT
OR 2 x "A".
SO FROM THIS ALONE SO LOOKING AT THE FORM OF OUR FUNCTION
IF P OF 15 MUST = 2A
WE WOULD HAVE "A" x THE BASE WHICH WE'LL LET = 2.
NOW WE CAN'T JUST RAISE THIS TO THE POWER OF T
BECAUSE WE KNOW THIS FUNCTION VALUE MUST EQUAL 2A.
THE EXPONENT WILL ACTUALLY HAVE TO BE T + DIVIDED BY 15
IN ORDER FOR P OF 15 TO EQUAL 2A.
NOTICE WHEN T IS 15,
THE EXPONENT ON 2 WOULD BE 15 DIVIDED BY 15 OR 1
AND "A" x 2 TO THE 1st IS 2A WHICH IS THE REQUIREMENT
BASED UPON THE GIVEN INFORMATION.
SO WE KNOW THAT OUR FUNCTION P OF T MUST BE IN THE FORM
OF "A" x 2 RAISED TO THE POWER OF T DIVIDED BY 15,
AND NOW SINCE WE KNOW THAT AFTER 75 MINUTES THE POPULATION
IS 72,000 WE KNOW THAT P OF 75 MUST EQUAL 72,000,
AND WE CAN USE THIS TO DETERMINE THE VALUE OF "A"
THE INITIAL POPULATION.
MEANING 72,000 MUST EQUAL "A"
x 2 RAISED TO THE POWER OF 75 DIVIDED BY 15.
WELL THIS WOULD GIVE US 72,000 = "A" x 2 TO THE--
75 DIVIDED BY 15 IS 5 SO NOW WE CAN SOLVE FOR "A"
BY DIVIDING BOTH SIDES BY 2 TO THE 5th OR 32.
THAT'S 72,000 DIVIDED BY 32
SO THE INITIAL AMOUNT OR THE INITIAL POPULATION
OF THE BACTERIA IS 2,250.
THIS ALSO GIVES US THAT P OF T MUST EQUAL 2,250 x 2 RAISED
TO THE POWER OF T DIVIDED BY 15.
SO TO FIND THE SIZE OF THE POPULATION
AFTER 4 HOURS WE JUST NEED TO CONVERT THE 4 HOURS TO MINUTES
AND THEN EVALUATE THE FUNCTION.
SO 4 HOURS IS = 4 x 60 MINUTES OR 240 MINUTES.
SO WE NEED TO FIND THE VALUE OF P OF 240
TO DETERMINE THE POPULATION AFTER 4 HOURS.
WELL 240 DIVIDED BY 15 = 16.
SO WE HAVE 2,250 x 2 TO THE 16th.
SO WE HAVE 147,456,000 BACTERIA AFTER 4 HOURS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.