Rose Garden Walking Path Addition

Question

a rose garden measures 12 feet by 20 feet. a landscaper is placing a walking path of uniform width around the rose garden. the area of the walking path is 68 square feet. How wide will the landscaper make the walking path?

Answer 1

Draw a rectangle inside a rectangle.

The inside rectangle has dimensions 12 ft by 20 ft

 

The dimensions of the outside rectangle are

(12+2x) by (20 + 2x)

 

We are looking solely at the area of the walkway

around the rose garden. 

 

         _________________________

        |   | ___________________|    |

        |   |                                   |    |

        |   |                                   |    |

        |   | 12                              |    | 12+2x

        |   |                  20              |    |

        |   |__ _________________|    | 

        |_ | ___________________|__|

          x                20                    x

 

Divide the outside pathway into 4 rectangles.

The way I did it created 2 rectangles that

are 20 by x and 2 rectangles that are

(12+2x) by x

 

 

The area of the walkway is the sum of the

areas of the 4 rectangles.

The area of a rectangle is length times width:

 

2 rectangles at area 20x

2 rectangles at area  x(12+2x)

 

The sum of the area of the 4 rectangles is 68

 

20x + 20x + x(12+2x) + x(12+2x)=68

40x + 12x + 2x² + 12x + 2x² = 68

 

combine like terms and put in descending order of exponent

 

4x² + 64x = 68

Make a quadratic by subtracting 68 from both sides

4x² + 64x - 68 = 0

Each term is divisible by 4 so factor out a 4

4(x² + 16x - 17) = 0

Divide both sides by 4

x² + 16x - 17 = 0

 

The factors of -17 that add to 16 are (17)(-1) so

this factors to

 

(x+17)(x-1) = 0

 

Either x+17 = 0 and x = -17 

OR

         x-1 = 0  and x = 1

 

Since the width will not be a negative, discard

the x=-17.

 

The pathway is 1 ft wide.