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# Size of a field

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Following the bounary of a rectangular field, it requires 11 rounds of a lawn mower to cut 1/2 of it and 14 more to cut the remainder. If the lawn mower cuts a swath 2 feet wide and all swaths are full, what are the dimensions of the field. Thank you for any help.

Feb 13, 2020

#1
+109561
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Since each pass of the  mower cuts a 2 foot swath.....the length  of the field  must  be

11*2 *2  + 14 *2* 2  =  (25)*4  =  100  ft = L

On the first  pass the square footage cut =

2* (W) * 2   + 2 (L - 4)*2  =

4W + 4(96)  =

4 (W + 96) =

4W  + 384

On the 2nd pass  the square footage cut =

2 (W - 4)*2  +2(L -8)*2 =

4W - 16 + 4(92) =

4(W + 76)=

4W  + 352=

4W + 384 - 32 (1)

On the 3rd pass the square footage cut =

2(W - 8)*2 + 4(L - 12) =

4W - 32 + 4 * 88 =

4W + 320 =

4W + 384 - 32(2)

.....

So.....on the first 11 passes....the square footage cut  is

4W   +  sum [ 384 - 32n] for n = 0 to n =10

11*4W + 2464

44W + 2464

.....

For the remaining 14 passes....

On the 11th  pass the square footage cut =

4W  - 32 (11)

......

On the 24th pass the square footage cut  =

4W  - 32 (24)

So....on the final 14 passes the square footage cut  =

14*4W  + sum [ 384 - 32n] for n = 11 to n =24

56W - 2464

And the  square footage of the first 11 passes = square footage of the remaining 14 passes...so.....

44W + 2464  = 56W - 2464   simplify

12W  = 4928

W =  [410 + 2/3 ] ft

Feb 13, 2020
#2
+22085
+1

I started' cutting'  down the same path to the solution as you......it seemed like a lot of work..... so I quit.

5 thumbs up !

Feb 13, 2020
#3
+109561
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I have a feeling that I missed something  .....

I just gave it my best shot   !!!!

CPhill  Feb 13, 2020
edited by CPhill  Feb 13, 2020
#4
+1

Since it requires 11 + 14 =25 rounds to cut the field and each round consists of 2 swaths, each 2 feet wide, then the width of the field is: 25 x 2 x 2 = 100 feet.

The 11 rounds represent 22 swaths of of 2 feet each, or 22 x 2 =44 feet. Hence, after 11 rounds, the width of the area remaining is: 100 - 44 =56 feet.

Now, if L represents the length of the field, then: (L - 44) x 56 =the remaining area. But, the remaining area =1/2 of the total area, which =1/2 x L x 100 =50L.

Therefore: (L -44) x 56 =50L, or 6L =2,464. And L =2,464/6 =410 2/3 feet - the length of the field.
So, the field =100 x 410 2/3 =41,066 2/3 ft^2.

Feb 13, 2020
edited by Guest  Feb 13, 2020
#5
+109561
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Thx,  Guest.....your procedure is WAY more elegant than mine

But....we  both got to the same destination

You took a jet

I took Greyhound

CPhill  Feb 13, 2020
edited by CPhill  Feb 13, 2020
#6
+1

Thanks CPhill. For every jet I take, you take at least 100 !!.

Feb 13, 2020
#7
+109561
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LOL!!!!

CPhill  Feb 13, 2020