Perimeter = 2(a+b) where a and b the width and the lengths of the rectangular garden.

Given: Perimeter = 60

60 = 2(a+b)

30=a+b --> solve for a variable, for no apparent reason, i'll solve for a, but it doesn't matter which one you solve for. 

30-b = a

area of the rectangular garden is a*b because a and b the width and the length of the rectangle

we know the value of a which is 30-b 

Area = a * b

Area = (30-b) * b

Area = 30b-b^2

We know this is a quadratric equation and in order to find the maximum area, we need to find the value of axis of symmetry 

Axis of symmetry is basically the value where Area is at its maximum position. 

Formula for axis of symmetry is-b/2a where in our equation b is 30 and a is -1. 

x = -30/2(-1) = 15

so now that we have the value of axis of symmetry, we plug this into our equation Area = 30b-b^2

Area = 30(15) - 15^2

Area = 450 - 225

Area = 225

Our maximum area is going to be 225. 

Now that we know our maximum area, we an find the dimensions pretty easily. 

a*b = 225 --> equation 1 (Area formula)

2(a+b) = 60 --> equation 2 (Perimeter formula)

Lets solve this using systematic equations. 

Lets rearrange equation 2 and solve for a variable, in this case i'll solve for b, but it doesn't matter which one you solve for. 

2(a+b) = 60

a+b = 30

b = 30-a

Now that we know the value of b which is 30-a, lets plug it into our area formula

a*b = 225

a (30-a) = 225

30a-a^2 = 225

a^2-30a+225=0 

use our quadratic equation

-b+/- sq (b^2-4ac)/2a

a= 1

b:=-30

c= 225

-(-30) +/- sq (30^2- 4(1)(225) / 2(1)

30+/- sq(0)/2

30+0/2 and 30-0/2

either ways, our solution is 15. 

so the value of a is 15. 

now that we know the value of a, which is 15, the other side dimension should be easy to figure out

a*b = 225

15*b = 225

b=15. 

so, the values of our dimensions are 15 by 15 and our area is 225. 

to check this, 15+15+15+15 = 60, which is our perimeter and 15*15 = 225 which is our maximum area. 

ANSWER: Dimensions: 15 ft by 15 ft

Maximum Area = 225 ft^2