Alegebra question...?

A developer wants to erect a rectangular


building on a triangular-shaped piece of property that is 200 feet


wide and 400 feet long (see the figure).





http://i7.photobucket.com/albums/y294/jo...





(A) Express the area A(w) of the footprint of the building as a


function of the width w and state the domain of this function.


[Hint: Use properties of similar triangles (Appendix B) to find a


relationship between the length l and width w.]





(B) Building codes require that this building have a footprint of


at least 15,000 square feet. What are the widths of the building


that will satisfy the building codes?





(C) Can the developer construct a building with a footprint of


25,000 square feet? What is the maximum area of the footprint


of a building constructed in this manner?





i need help!

Alegebra question...?
(A)


put lower left corner at (0,0), and upper right corner is (x,y).





y= -1/2 x +200 (y = mx +b form for equation of line, which you can tell by inspection)





Area of building = x times y





Area = -1/2 x^2 + 200x, but x = width, so


A(w) = -1/2w^2+200w


domain are the valid values for w, so it is from 0 to 400.





(B)


Set A(w) = 15,000 and solve for w





15,000 = -1/2w^2 + 200w





0.5w^2 - 200w + 15,000 = 0, now multiply by 2





w^2 - 400w + 30,000 = 0





Notice 300 x 100 = 30,000 and -300 and -100 make -400





So factoring the quadratic equation,





(w - 100) x (w - 300) = 0





So clearly, w = 100 and w = 300 satisfy this equation.





(C) Finding the maximum area.





Find first derivative of A(w), and set it = 0 to find maximum area.





-w + 200 = 0, w=200 gives maximum area.


A(w) = -0.5(200)^2+200(200) =20,000. So answer to (c) is NO, 20,000 sq ft is the largest.