calculus questio?

Question

1.Suppose the number of houses in a new subdivision after t months of development is modeled by N (t) = 1000t3 /100 + t3 ,
where N is the number of houses and t >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

Do you mean N(t) = 1000*t^3 / (100 + t^3)?

The way you wrote it would mean N would tend to infinite if t tends to infinite and there would be no way to answer a)

So a) is what is lim N(t) where t->infinity

For this you would use Hopitals Rule where

lim a(t)/b(t) = da(t)/dt / db(t) / dt

so in this case lim N(t) = 1000 * d(t^3)/dt / d(100+t^3)/dt

= 1000 * 3 t^2 / 3*t^2 = 1000

so lim N(t) as t approachs infinity = 1000

b) infinity as the function asymptotically approachs 1000 as t rise - i.e there is no maxima

Answer 2

Please use appropriate bracketing to make your question easier to read!

I assume you mean
N(t) = 1000 t^3 / (100 + t^3)
= 1000 (100 + t^3) / (100 + t^3) - 1000 (100) / (100 + t^3)
= 1000 - 100000 / (100 + t^3)

N(t) is thus a monotonic increasing function for t>0. As t-> infinity N(t) -> 1000, so presumably this is the answer they want for (a), but this would imply that the subdivision is never fully developed, so it's rather an odd question. For instance, after 10 years N(t) = N(120) = 999.94 houses... at what point do we say the last house is complete? Or do we say the subdivision is complete at 999 houses? This will happen when 100 + t^3 = 100000, or t^3 = 99900 => t = 46.4 months (1 dp) or 3 years 10.4 months, and that last house just keeps on getting tinkered with whenever someone has time. ;-)