Jessica A.
asked • 12/23/16HELPPPPPPP
A pool measuring 12 meters by 24 meters is surrounded by a path of uniform width, as shown in the figure. If the area of the pool and the path combined is 640 square meters, what is the width of the path?
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2 Answers By Expert Tutors
Kenneth S. answered • 12/23/16
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Expert Help in Algebra/Trig/(Pre)calculus to Guarantee Success in 2018
If x is the path's width, then the overall area is (12+2x)(24+2x) = 640 sq m.
Solve that quadratic. Hint: divide each factor by 2 and divide 640 by 4 to simplify:
(6+x)(12+x) = 160. Solve THIS quadratic (FOIL left side & then put all terms on left side--if lucky, you can solve by factoring).
Kenneth S.
Perhaps the suggestion to divide each factor (there are two parenthesized binomial factors on the left side) by two was not understood as equivalent to dividing the left side by four (which I wanted done to the right side of my first equation).
So perhaps that's why there was a downvote. Or perhaps because student would only be satisfied with a complete solution, down to x = a specific number, thus handing the answer to her. But I don't think we should be handing out answers too often--we best serve the students by helping them set up the problem and inviting them to (a) think (b) apply basic algebra and (c) check the answer(s).
Oh, well, Merry Christmas to all.
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12/23/16
Andrew M. answered • 12/24/16
Tutor
New to Wyzant
Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors
When you state "the area of the pool and path combined is 640 square meters"
do you mean the sum of the area of the pool and the area of the path is 640 m2 or that
the total as a single rectangle has an area of 640 m2 ?
Kenneth showed how to set it up as a single rectangular area of dimensions
(24+2x) by (12+ 2x), which is fine if that is what was intended.
Otherwise: We have 24X12 = 288 m2 covered by the pool.
640 -288 = 352 m2 of area encompassed by the path around it.
That area is actually 4 rectangles whose areas sum to 352 m2
Those can be set up as either 2 rectangles at (12+2x) by x
and 2 rectangles of 24 by x
OR
Two rectangles of (24+2x) by x and two of 12 by x
***********************************************
When I drew it out I labeled it as 4 rectangles that were
composed of 2 with dimensions 24 by x and 2 with dimensions (12+2x) by x
For a 12+2x by x rectangle the area is:
x(12+2x) = 12x + 2x2 of which there are 2 so...
2(12x + 2x2) = 24x + 4x2
For the 24 by x area the area is 24x
There are also 2 of these so we have 48x
4x2 + 24x + 48x = 352
4x2 + 72x = 352
4x2 + 72x - 352 = 0
As Kenneth noted before, we can reduce this quadratic
by factoring out a 4.
4(x2 + 18x - 88) = 0
x2 + 18x - 88 = 0
Solve this quadratic to find the width of the path.
Note this is factorable so the quadratic formula isn't really necessary.
When you find the two possible answers, discard the negative.
The other answer will be positive and will be the width in meters of the path.
Kenneth S.
Andrew, in his recent answer, wrote: When you state "the area of the pool and path combined is 640 square meters"
do you mean the sum of the area of the pool and the area of the path is 640 m2 or that
the total as a single rectangle has an area of 640 m2 ?
do you mean the sum of the area of the pool and the area of the path is 640 m2 or that
the total as a single rectangle has an area of 640 m2 ?
TOTAL has a single meaning! There is no doubt here, and the expression that I wrote to reflect said total area contains, embedded, the pool's area, IMPLICITLY, and the area of the uniform width (x) path along its four sides.
This is because the overall combined area is rectangular, and the pool's smaller rectangular area is part of the overall rectangle.
This is because the overall combined area is rectangular, and the pool's smaller rectangular area is part of the overall rectangle.
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12/24/16
Andrew M.
When you're right, you're right Kenneth.
I did that at the end of a long day. The
actual work is valid and the answer is right,
but an unnecessary and too complicated
method. Merry Christmas.
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12/25/16
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12/24/16