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

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: