Theresa won a gardening contest and was awarded a roll of deer-proof fencing. The fencing is 36 feet long. She and her husband, John, discuss how to best use the fencing to make a rectangular garden. They agree that they should only use whole numbers of feet for the length and width of the garden.

a. What are all of the possible dimensions of the garden?

b. Which plan yields the maximum area for the garden? Which plan yields the minimum area?

see below

Step-by-step explanation:

It is given that the roll of fencing is 36 ft long. Hence the perimeter of the fencing is 36 ft long.

Since the fence is to be rectangular,

Perimeter = (2 x Length) + (2 x Width) = 36 feet

The quickest would be using calculus, but the most simple way (without using polynomials) would be to simply list the possibilities for Length and Width:

If we consider only half of the perimeter, i. e only 1 Length and 1 Width:

From the formula above,

(1/2) of Perimeter = Length + Width = 18 feet

Given that the Length & Width are to be whole numbers AND that the sum of Length & Width adds up to 18, then the possible dimensions and area are as follows:

Assemble a table in the following format:

Length x Width = Area

18 x 1 = 18

17 x 2 = 34

16 x 3 = 48

15 x 4 = 60

14 x 5 = 70

13 x 6 = 78

12 x 7 = 84

11 x 8 = 88

10 x 9 = 90

9 x 10 = 90 (anything from here onward basically repeats the previous lines, so we ignore this line and any others that follow)

a) The possible dimensions are everything listed in the table above except the last line which is repeated

b) we can see plainly that the maximum area is 10ft x 9ft = 90 sq feetand the minimum area is 18ft x 1ft = 18 sq feet