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.
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
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 !
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.