SOLUTION: A herd of deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. It can be shown by means o

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Question 842477: A herd of deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. It can be shown by means of calculus that the rate R (in deer per year) at which the deer population changes at time t is given by R = −4t3 + 38t.
(a) When does the population cease to grow? (Round your answer to two decimal places.)
after t = Incorrect: Your answer is incorrect. years
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Same as asking find t for the derivative of R equal to zero.
(*See note below)

dR%2F%28dt%29=-12t%5E2%2B38
-
Find solution for t, for the one such that t%3E0.
-12t%5E2%2B38=0
12t%5E2=38
6t%5E2=19
highlight%28t=sqrt%2819%2F6%29%29 years
-
This time quantity is 1.78 years, or 1 year 41 weeks.



* assumed your equation was to be R=-4t%5Ehighlight%283%29%2B38t

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the formula is:
R = -4t^3 + 38t.
The point in time at which the population of the herd stops to grow would be at the maximum growth which is when the curve changes from increasing per year to decreasing per year.
the slope of the curve at that point in time would be equal to 0.
you can use calculus to find this point.
take the derivative of the equation which will give you the equation of the slope and set that equation equal to 0 and you will find the points where the rate is 0.
the maximum or minimum point will be at those points.
to graph your equation, replace R with y and replace T with x to get:
y = -4x^3 + 38x
take the derivative of this equation to get:
y' = -12x^2 + 38
set this equal to 0 to get:
-12x^2 + 38 = 0
add 12x^2 to both sides of this equation to get:
12x^2 = 38
divide both sides of this equation by 12 to get:
x^2 = 38/12
take the square root of both sides of this equation to get:
x = +/- sqrt(38/12)
your solution should be:
x =+/- 1.779513042.
since x can't be negative, the solution has to be x = 1.779513042.
i checked it out on my ti-84 calculator and that solution turns out to be correct.
the graph of your original equation of y = -4x^2 + 38x is shown below:
graph%28600%2C600%2C-1%2C5%2C-10%2C50%2C-4x%5E3%2B38x%29
you can see from the graph that the rate of growth maxes out at x = 1.78 rounded to 2 decimal places.