Alandscaper wants to create a 12-foot-long diagonal path through a rectangular garden. the width of the garden is x feet and the length of the garden is 4 more than the width. he uses the pythagorean theorem to write an equation to determine the width of the garden. (x)2 + (x + 4)2 = (12)2 x2 + x2 + 8x + 16 = 144 2x2 + 8x – 128 = 0 what are the approximate dimensions of the garden? 6.2 ft by 2.2 ft 6.2 ft by 10.2 ft 10.2 ft by 2.2 ft

Dimensions of the gardens are 6.2 ft by 10.2 ft .

Step-by-step explanation:

As given

A landscaper wants to create a 12-foot-long diagonal path through a rectangular garden.

The width of the garden is x feet and the length of the garden is 4 more than the width.

Length of the garden = x

Breadth of the garden = x + 4

By using the pythagorean theorem

Hypotenuse² = Perpendicular² + Base²

12² = x² + (x + 4)²

As (a+ b)² = a² + b² + 2ab

As 12² = 144

144 = x² + x² + 16 + 2 × x × 4

144 = 2x² + 16 + 8x

2x² + 8x + 16 - 144 = 0

2x² + 8x - 128 = 0

x² + 4x - 64 = 0

As the general  form of the equation is in the form

ax² + bx + c = 0

a = 1 , b = 4 , c = -64

Now by using the discriminant formula

Thus

x = 6.2 ft (Approx)

x = -10.2 ft (Approx)

(As the sides of the rectangular garden cannot be negative .)

Thus

x = 6.2 ft

Length of the garden = 6.2 ft

Breadth of the garden =  x + 4

                                     = 6.2 + 4

                                     = 10.2 ft

Therefore the dimensions of the gardens are 6.2 ft by 10.2 ft .

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2

answer: 2530

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