Number of houses in a subdivision after t months of development is modelled by Equation N(t) =1000t^3/ 100+t^3 where N is number of houses and t greater than equal to zero. How many houses can be expected to be in the subdivision when the development is complete?
The building development represents an area that must be fully completed in a interval of time.
The subdivision of development represents a small area of the development.
A completed building development comprises the maximum number of houses in each subdivision.
Since the function `N(t) = (1000t^3)/(100+t^3)` , that models the number of houses in each subdivision, needs to be maximized, you need to solve for t the following equation, such that:
`N'(t) = 0`
Differentiating the function N(t) with respect to t, using the quotient rule, yields:
`N'(t) = ((1000t^3)'(100+t^3) - (1000t^3)(100+t^3)')/((100+t^3)^2)`
`N'(t) = (3000t^2(100+t^3) -(1000t^3)*(3t^2))/((100+t^3)^2)`
`N'(t) = (300000t^2 + 3000t^5 - 3000t^5)/((100+t^3)^2)`
Reducing duplicate terms yields:
`N'(t) = (300000t^2)/((100+t^3)^2)`
Since` N'(t) = 0` if `300000t^2 = 0` yields:
`300000t^2 = 0 => t^2 = 0 => t = 0`
Hence, the function N(t) is maximized at t = 0.