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- A POPULATION OF BEETLES
ARE GROWING ACCORDING TO A LINEAR GROWTH MODEL.
THE INITIAL POPULATION IS P SUB 0 = 7,
WHICH MEANS AT WEEK 0 THE POPULATION WAS 7
AND THE POPULATION AFTER 7 WEEKS IS P SUB 7 = 56.
WE WANT TO FIND THE EXPLICIT FORMULA
FOR THE BEETLE POPULATION AFTER N WEEKS
AND ALSO AFTER HOW MANY WEEKS
WILL THE BEETLE POPULATION REACH 147.
THE EXPLICIT EQUATION WE WANT TO FIND IS GIVEN HERE BELOW.
P SUB N = P SUB 0 + D x N.
WHERE P SUB 0 IS THE INITIAL POPULATION WHICH WE KNOW IS 7,
AND THE COMMON DIFFERENCE, D,
IS THE SAME AS THE SLOPE OF THE LINE
WHICH WOULD BE EQUAL TO THE CHANGE IN THE POPULATION
DIVIDED BY THE CHANGE IN TIME.
LET'S BEGIN BY RUNNING THE GIVEN INFORMATION AS ORDERED PAIRS.
SO FOR P SUB 0 = 7, WE WOULD HAVE THE ORDERED PAIR AS (0,7)
WHERE THE FIRST COORDINATE IS THE NUMBER OF WEEKS
AND THE SECOND COORDINATE WOULD BE THE POPULATION.
AND SINCE P OF 7 = 56, THAT WOULD BE THE ORDERED PAIR (7,56)
AND NOW WE CAN FIND THE COMMON DIFFERENCE D,
WHICH IN THIS CASE WOULD BE THE CHANGE IN THE POPULATION
DIVIDED BY THE CHANGE IN TIME.
SO THE COMMON DIFFERENCE D IS = TO THE DIFFERENCE IN P
DIVIDED BY THE DIFFERENCE IN W.
SO WE'D HAVE 56 - 7 DIVIDED BY 7 - 0,
WHICH WOULD BE 49 DIVIDED BY 7 WHICH = 7.
THIS REPRESENTS THE COMMON DIFFERENCE
WHICH WOULD ALSO BE THE LINEAR RATE OF CHANGE PER WEEK.
THIS TELLS US THE POPULATION HAS INCREASING 7 PER WEEK.
AND NOW FOR OUR EXPLICIT EQUATION
WE WOULD HAVE P SUB N = P SUB 0 WHICH IS 7 + (D x N),
WE KNOW D IS 7, SO WE HAVE + (7,N).
NOW WE WANT TO KNOW AFTER HOW MANY WEEKS
WILL THE BEETLE POPULATION REACH 147.
NOTICE THIS 147 IS A POPULATION NOT THE NUMBER OF WEEKS.
SO THIS IS TELLING US THAT P SUB N = 147.
WE WANT TO SOLVE FOR N.
SO WE WANT TO SOLVE THE EQUATION, 147 = 7 + (7,N)
SO WE'LL SUBTRACT 7 ON BOTH SIDES.
SO WE HAVE 140 = (7,N).
NOW DIVIDE BOTH SIDES BY 7, 140 DIVIDED BY 7 = 20.
SO WE HAVE 20 = N.
SO TO ANSWER THE QUESTION,
AFTER 20 WEEKS, THE POPULATION WILL REACH 147.
I HOPE YOU FOUND THIS HELPFUL.