The population of a small town is modelled by the function:?

Question

P(t)=10 (4t+3)/2t+5 , where P(t) is the population in thousands, and t is time in years since the start if 1990

a. is the population currently increasing or decreasing

b. The town will need its own transit system if the population excess 50,000. Will the town's population ever exceed 50 000? Explain.

Sqdancefan · Answer

Assuming your function is
.. P(t) = 10(4t+3)/(2t+5)
it can be rewritten as
.. P(t) = 20 - 70/(2t+5)

a) The function P(t) is increasing for all values of t. P(t) is currently increasing.
.. P'(t) = 140/(2t+5)^2 ... so cannot be negative

b) The function P(t) has a horizontal asymptote at P(t) = 20. The value of the function will never exceed 20, so will not reach 50.

Wayne DeguMan · Answer

We require P '(t) as we require the rate of change function to determine and increase or decrease.

Using the 'quotient rule' we get:

P '(t) = 140/(2t + 5)²

Regardless of the value of t, (2t + 5)² > 0...hence, P '(t) > 0...i.e. increasing

Now, P(t) = 20 - 70/(2t + 5)

As t --> ∞, 70/(2t + 5) --> 0...so, P(t) --> 20

Hence, the population will approach 20,000....so, never 50,000!!

:)>

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