A contractor has 34 meters of fencing that he is going to use as the perimeter of a rectangular garden. The length of one side of the garden is represented by x, and the area of the garden is 72 square meters. Determine algebraically the dimensions of the garden. show work for full credit.

Given

Shape of garden: Rectangle

Perimeter: 34 meters

Area: 72 square meters

Length of one side = x

Required

Find the dimension of the rectangle

Let the length of the other side of the rectangle be represented with y.

We start by writing out the perimeter of a rectangle

Perimeter, P = 2(L + B)

Where P is the perimeter; L and B are the length and width of the sides of the rectangle.

In this case L and B are represented by x and y

So,

P = 2(x + y)

By substituting 34 for P

34 = 2(x + y)

Multiply both sides by ½

½ * 34 = ½ * 2(x + y)

17 = x + y

Make y the subject of formula

y = 17 - x

Recall that the sides of the rectangle are represented by x and y.

Hence, the area of the rectangle is as follows.

Area = x * y

Substitute 72 for Area and 17 - x for y

72 = x(17 - x) --- Open bracket

72 = 17x - x² Reorder

x² - 17x + 72 = 0

Now, we have a quadratic equation

Solving by factorisation

x² - 8x - 9x + 72 = 0

x(x - 8) - 9(x - 8) = 0

(x - 8)(x - 9) = 0

x - 8 = 0 or x - 9 = 0

x = 8 or x = 9

Recall that

y = 17 - x.

When x = 8

y = 17 - 8

y = 9

When x = 9

y = 17 - 9

y = 8

Writing out the result, we have

x = 8 , y = 9

Or

x = 9, y = 8

Hence, the dimension of the rectangle is 8 metres by 9 metres

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