U3.L4.HW: Quadratic Functions: Question 24

Question

Put you answers first, then explain them in the end.

Cornell wants to build a rectangular enclosure for his animals. One side of the pen will be against the barn, so he needs no fence on that side. The other three sides will be enclosed with wire fencing. If Cornell has 450 feet of fencing, you can find the dimensions that maximize the area of the enclosure.

a) Let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). Write an function for the area A of the enclosure in terms of w. (HINT first write two equations with w and l and A. Solve for l in one equation and substitute for l in the other).

A(w) = ________

b) What width w would maximize the area?

w = ____________ feet

c) What is the maximum area?

A = ____________ square feet

Answer 1

450 = L+2w

L = 450-2w

A =Lw = (450-2w)w = 450w -2w^2

A' = 450 -4w = 0

4w =450

w =450/4 = 112.5 feet wide

L = 450-2(112.5) = 450=225 = 225 feet Long

225 ft by 112.5 ft maximizes Area = 225x112.5

= 25,312.5 square feet max