If the mower cuts a three foot wide path, at what point should Erin stop mowing and Aimee start mowing?

Question

Erin and Aimee are each responsible for mowing half of their back yard. The yard is rectangular with dimensions 75 feet by 90 feet. Erin starts mowing at a corner, gradually working her way toward the middle with concentric bands around the outside edges. If the mower cuts a three foot wide path, at what point should Erin stop mowing and Aimee start mowing?

Answer 1

Hi Jamie,

I am going to assume that the blade of the mower has a radius R and when it rotates it sweeps a path that is 2R.

2R =3 ft 

L  = 90 feet

W = 75 feet

Since Aimee and Erin each mowing half of the yard, assuming that Erin is moving along the length that spans 90 ft from the corner where she starts mowing we can draw the following diagram.

 

Erin starts here

•------ .   .--------------------------------------------------------------•• Erin first run

*   .          .                                                                            *  N = 1

*  .            . ----->                                                                 2R = 3 ft

*   .          .                                                                            *

*      .    .                                                                               *

* -----------------------------------------------------------.   .---------*

*                                                                       .          .        *

*                                                           <----    .            .       *       

*                                                                       .          .        2R = 3 ft

*                                                                           .   .           * 

*------------------------------------------------------------------------ *                        

*                                                                                           *

.                                                                                            .

.                                                                                            .

.                                                                                            .

 

 

 

*                                                                                           *

*                                                                                           *

*                                                                                           *

*                                                                                           * 

*                                L = 90 feet                                           *

*****************************************************•End Point 

The distance from N = 1 to End point is W = 75 feet

 

Now it is obvious from the sketch above that Erin needs to mow 'N' times before stopping.

If  N = 1,  W₁ = 2R = 3ft, L = 90 ft

If  N = 2,  W₂ = 2(2R) = 6 ft, L = 90 ft    

Find N such that  W_N =1/2 W = 75/2 = 37.5 ft, L = 90 ft

N(2R) = 37.5

N (3)  = 37.5

N = 37.5/3 = 12.5

Erin should stop after mowing throught 12 1/2 times the length of the yard each time covering an area equal to

A = 90ft x3ft = 270 ft².

Answer 2

After traveling 1125 feet.  

 

75x90 = .75x 9000 = (3/4)9000

 

3(9000/4) = 3(2250)

 

so a 3 foot swath will involve mowing in a line 2250 ft long.  After half that they should swap out.