Calculus question......plz answer it in detail?

Question

suppose the number of houses in a new subdivision after t months of development is modelled by N(t) = 1000t power 3 divide
100 + t power 3,

where N is the number of houses and and t is greater thn 0.
a) how many houses will be in the subdivision when the development is complete?
b) after how many years is the subdivision fully developed ?

Answer 1

I interpret part a) to mean what's the upper limit to the number of houses? So I consider the limiting value of N(t) as t increases. If we divide both top and bottom by t^3, we find it approaches
1000/(0 + 1), which is 1000.

I need to check that N(t) is increasing:

N'(t) = (300000t^2)/(100+t^2)^2 -- not sure about this, did in a hurry without checking. But it's positive so N(t) is increasing. So answer to (a) is 1000.

Now to answer b), I'd find the value of t at which N(t) exceeds 999.5, and take that as the month in which full development is achieved.
i.e. solve

(t^3)/(100 + t^3) > 0.9995
This gives
t^3 > 199900 and so
t > 58.47, so development is complete after 59 months, or almost 5 years.

Answer 2

a) All you have to do is plug in whatever your t is into the formula. For example, if t = 3, then you would plug it into the formula like this: N(3) = 1000(3)^3/100(3^3) and solve.
b) What is considered fully developed? Is it 1000 houses, 100000 houses, or 3 houses?

Answer 3

You have a relation for N(t) which means for a given t you can find N and for a given N you can probably find t.

Niether a) or b) are answerable since you haven't defined the fully developed values for t or N

Answer 4

You can solve this model, but I believe you have made a mistake in representing the model, as it doesn't behave very nicely.


Hy (below) interpreted the function differently than me. His function does converge, and I agree with his analysis. However, the model is really not a good model of the building process.