Question

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. 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

Answer 1

Dimensions of the gardens are 6.2 ft by 10.2 ft .

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

`x=(-b\pm\sqrt(b^(2)-4ac))/(2a)`

`x=(-4\pm\sqrt(4^(2)-4* 1* -64))/(2* 1)`

`x=(-4\pm\sqrt(16+256))/(2)`

`x=(-4\pm\sqrt(272))/(2)`

Thus

`x=(-4+\sqrt(272))/(2)`

`√(272) = 16.49`

`x=(-4+16.49)/(2)`

`x=(12.49)/(2)`

x = 6.2 ft (Approx)

`x=(-4-\sqrt(272))/(2)`

`√(272) = 16.49`

`x=(-4-16.49)/(2)`

`x=(-20.49)/(2)`

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 .

Answer 2

If we let x be the width of the garden, then x+4 will be the length. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs.

12² = (x)² + (x + 4)²

Simplifying,

144 = x² + x² + 8x + 16

Further simplification,

144 = 2x² + 8x + 16

x² + 4x - 128 = 0

The value of x from the equation is 6.2 ft. The length is then equal to 10.2 ft.


The answer to this item is the second choice.