Answer:
length of the pretty side and length of the side oppositte to the pretty side = 37.91 ft
length of the other two sides = 27.52 ft
Step-by-step explanation:
The mathematical problem is:
Max A = b1*h
subject to: 35*b1 + 18*(2*h + b2) <= 3000
Where
A: area of the garden
b1: length of the pretty side
b2: length of the side oppositte to the pretty side
h: length of the other two sides
Replacing with b1 = b2 and taking only the equality sign in the restriction (in the maximum all the money will be spent), we get:
35*b1 + 18*(2*h + b1) = 3000
35*b1 + 36*h + 18*b1 = 3000
53*b1 + 36*h = 3000
b1 = 3000/53 - (36/53)*h
Substituing in Area's formula
A = (3000/53 - (36/53)*h)*h
A = (3000/53)*h - (36/53)*h^2
In the maximum, the derivative of A is equal to zero
dA/dh = 3000/53 - 2*(36/53)*h =
3000/53 - 72/35*h = 0
h = (3000/53)*(35/72)
h = 27.52 ft
then,
b1 = 3000/53 - (36/53)*27.52
b1 = 37.91 ft =b2
