The local school board wishes to build a new school and they

The local school board wishes to build a new school, and they are trying to figure out a design for the adjacent playground. The rectangular playground will be built adjacent to the school by building three fences. Two of the fences will be built out of standard fencing which costs $30 per meter, while the third, the one adjacent to the road, will be built out of heavy duty fencing which costs $60 per meter. If the total budget for the fence is $2700, find the dimensions of the playground with largest area that can be built. Make sure to justify your answer algebraically using known facts about the maximum or minimum of a quadratic function. Do not use calculus to solve this problem as you “should” not know calculus yet.

Solution

P= x+2y

Cost = 60x+2*y*30

2700=60x+60y

45=x+y

y= 45-x

A=X*Y

A= X(45-X)

A= -x^2 +45x

we will use vertex

X= -b/2a= -45/-2 = 45/2

Y= 45-x = 45-45/2 = 45/2

So length and width = 45/2 which will give maximum value