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Question 1138875: A zoologist tracks the population of a species of butterfly at the zoo over several years. From the table of values, determine an approximate equation that models the butterfly’s population growth and determine the number of years before the population is 5 times greater than the original number.
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Year(t) 0 1 2 3 4
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Population 22 29 37 48 63
P(t)
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Found 2 solutions by rothauserc, greenestamps:
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! mean of t is 10/2 = 5
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mean of P(t) is 199/5 = 39.8
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SStt = summation for i = 1, 5 of t(i)^2 - (1/5) * (summation for i = 1, 5 of t(i))^2 = 30 - 10^2/5 = 10
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SSpp = summation for i = 1 to 5 of P(i)^2 - (1/5) * (summation for i = 1, 5 of P(i))^2 = 8967 - 199^2/5 = 1046.8
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SStp = summation for i = 1, 5 of t(i)P(i) - (1/5) * (summation for i = 1 to 5 of t(i)) * (summation for i = 1 to 5 of P(i)) = 499 - 10 * 199/5 = 101
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slope = SStp/SStt = 101/10 = 10.1
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y-intercept = mean of P(t) - mean of t * slope = 39.8 - 2 * 10.1 = 19.6
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The equation of the regression line is
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P(t) = 10.1t + 19.6
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5 * 22 = 110
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110 = 10.1t + 19.6
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10.1t = 90.4
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t = 8.95 is approximately 9 years
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There are 9 years before the population is 5 times greater than the original number
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Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
The analysis by tutor @rothauserc is fine; and his answer is probably the answer expected by the author of the problem.
However, when the population is "5 times GREATER THAN" the original population, the grammatically correct interpretation is that the population is the original population, plus that population 5 times MORE. So when the population is 5 times greater than the original population of 22, it is 22 plus 5*22 = 6*32 = 132.
So the final calculations to get the GRAMMATICALLY CORRECT answer are...
6*22 = 132
132 = 10.1t+19.6
10.1t = 112.4
t = 11.13 (to a couple of decimal places)
So the population will be 5 times greater than the original population in about 11 years.
Be wary (or downright suspicious!) of any problem which says something like one number is "5 TIMES GREATER THAN" another number. Nearly always, the intended meaning is "5 TIMES AS LARGE AS".
But the two phrases mean different things.
5 times as great as 22 is 5*22 = 110.
5 times greater than 22 is 22 plus 5*22 = 132.
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