If the instantaneous rate of change of a population is #50t^2 - 100t^(3/2)# (measured in individuals per year) and the initial population is 25000 then what is the population after t years?

Answer 1
#P(t)= 50/3t^3-40t^(5/2)+25000#

In order to respond to this question, you must identify a specific antiderivative.

Let #P(t)# be the population #t# years after the initial population was counted. (So #P(0)=25000#)

The population is changing at a rate, which indicates that we will soon learn about the derivative, or the rate of change. Stop right there.

The population is changing at a rate of #50t^2-100t^(3/2)#.
This tells us that #P'(t)=50t^2-100t^(3/2)#.
Our task is to find #P(t)#
If #f'(x)=x^n#, the #f(x)=x^(n+1)/(n+1)#
So #P(t)=50t^3/3-100*2/5 t^(5/2)+C# . . (See below if needed)
#P(t)=50/3t^3-40 t^(5/2)+C#
We also know that #P(0)=25000#, so we can substitute to find #C#
#25000=50/3(0)^3-40 (0)^(5/2)+C#, so #C=25000# #P(t)= 50/3t^3-40t^(5/2)+25000#
I think writing all of the following clutters up the work: To find the antiderivative of #t^(3/2)#:
#t^(3/2+1)/(3/2+1)=t^(5/2)/(5/2)=2/5 t^(5/2)#

(Multiplying by the reciprocal is what divides.)

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Answer 2

The population after ( t ) years is given by the antiderivative of the rate of change function plus the initial population. Thus, the population function is:

[ P(t) = \int (50t^2 - 100t^{3/2}) dt + 25000 ]

[ P(t) = \frac{50}{3}t^3 - \frac{200}{5/2}t^{5/2} + 25000 + C ]

[ P(t) = \frac{50}{3}t^3 - \frac{400}{\sqrt{t}} + 25000 + C ]

Since the initial population is 25000, we can solve for ( C ):

[ P(0) = \frac{50}{3}(0)^3 - \frac{400}{\sqrt{0}} + 25000 + C = 25000 ]

[ 25000 = 0 - \frac{400}{0} + 25000 + C ]

[ 25000 = 25000 + C ]

[ C = 0 ]

Thus, the population function is:

[ P(t) = \frac{50}{3}t^3 - \frac{400}{\sqrt{t}} + 25000 ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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