Math Factoring and Quadratic Equations

Question

A rectangular garden has dimensions 40m by 26m. The city is planning to construct three pathways of equal width in the garden, two of which are parallel to the 26m side, and the third one is parallel to the 40m side. As a result, each piece of the subdivisions of the garden is 144m^2.

Determine the width of the pathways.

Answer 1

2(23-w)w + (40)w = 40(23) - 144

46w -2w^2 +40w = 920-144

86w -2w^2 = 776

2w^2 - 86w =-776

2w^2 - 86w - 776 =0

w^2 -43w + 388= 0

w= 43/2 +/- (1/2)sqr(43^2 + 4(388))

w = 21.5 +/- .5sqr(1849 + 1552)

= 21.5 +/-sqr(3401)/2

= 21.5 +/-58.32/2

= 21.5 +/- 29.16

= 50.66 or -7.66

which is impossible as w can't be negative or greater than 23

this all assumes I understand the question. It could be interpreted differently. Somewhat ambiguous. The above does capture the quadratic, but not any integer "factoring" Odds are a different interpretation that leads to integer factors might be the way to go. Or maybe there's an arithmetic mistake above somewhere. Is there a diagram that comes with this problem?

Maybe the 3 pathways in the garden intersect, not at the end corners, but fully inside the garden, then:

40w + (2)23w - 2w^2 = 40(23) -144

2w^2 - 86w + 920-144 = 0

2w^2 -86w + 776

w^2 -43w + 388= 0

w = 43/2 +/- .5sqr(43^2 - 4(388)

w = 21.5 +/- .5(1849-1552)

w= 21.5 +/-.5(297)

w=21.5 + or - 8.6

w =40.1 or 12.9 meters = pathway width

40.1>23 so it's impossible

still no integer factors though

Maybe the 144 m^2 subdivisions refer to the 6 different areas, created by the 3 intersecting pathways. IF so, then

40w + 2(23)w -2w^2 = 40(23) - 6(144)= 912-864=48

2w^2 -86w +48 = 0

w^2 -43w +24=0

w= 43/2 +/- .5sqr(43^2-4(24))

w=21.5 +/- .5sqr(1849-96)

w=21.5+/- .5sqr(1753)

w=21.5+/- 20.9

w= 42.4 or 0.6 meters

still no integer factors, 42.4 is impossible

Or maybe the pathways intersect inside the guarden, but just once inside not twice, creating 2 not 6 subdidivions. tthe 2nd intersection at one corner

then you get another solution: 33.6 or 9.4 meters. 33.6 is impossible,

still no integer factors though

Or they could intersect in a way that creates 3 or 4 subdivisions.

Half the problem is figuring out what the problem is.

with 3 subdivisions w=31.4 or 10.6

31.4 is impossible, and still no integer factors

with 4 subdivisions w =42.4 or 0.6

42.4 is impossibe and still no integer factors

subdivisions could be 1, 2,3,4 or 6

divided in paths with shapes like F and H, and extensions of parts of the F and H.

as it is possible values for w are approximately 0.6, 9.4, 10.6, or 12.9 meters

rounded off to nearest tenth of a meter

depending on number of subdivisions

(caveat: assuming no arithmetic errors in the above calculations. it gets tedious.)