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...

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? 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.

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