A rectangular sandbox has a width of W feet, a perimeter of P feet, an

Given A rectangular Sandbox, with Width = W, Perimeter = P & Area = A

Let Height = H & Length = L. we get

\(A = 2({LW} + {WH} + {LH})\)....(i)

\(P = 4(L+W+H)\)......(ii)

Multiply (ii) by W on both sides

\(WP = 4({LW}+W^2+{HW})\)

\(WP = 4({LW}+{HW}) + 4W^2\)

From (i), we have \(2({LW} + {WH}) = A - 2{LH}\)

Therefore \(WP = 2(A - 2{LH}) + 4W^2\)

\(4W^2 - WP + 2(A - 2{LH}) = 0\)

Now, in devoid of any information about the Length & Height of the Sandbox, (not sure whether its a faulty question or am i missing something)
to obtain an expression from among the answer choices, requires an assumption , \(W=2L=H\), hence \(2LH = W^2\)


We get \(4W^2 - WP + 2(A - W^2) = 0\)

\(2W^2 - WP + 2A = 0\)

Answer E.

Thanks,
GyM

We can assume simple values for all the variable in the regular sandbox

Width(W) = \(4\) ft | Perimeter(P = 20) -> \(2(L + W) = 20\) -> \(L = 6\) | Area(A) = LW = \(24\) sq ft.

Substituting the values of W, P, and A in all the answer options
a. \(W^2 + PW + A = 0\) Cannot be zero.
b \(W^2 - PW + A = 0\) -> \(16 - 80 + 24 = -40\)
c. \(2W^2 + PW + 2A = 0\) -> Cannot be zero
d. \(2W^2 - PW - 2A = 0\) -> 2*16 - 80 - 48 = -96
e. \(2W^2 - PW + 2A = 0\) -> 2*16 - 80 + 48 = 0 (Option E)

GyMrAT - Unfortunately, the calculations you have done are wrong(because Area is in Square feet)

pushpitkc Not sure where i have gone wrong, can you point out the error. Also the question says its a "Rectangular Sandbox", i guess that means its a 3D figure, please correct me if i am wrong.


Thanks,
GyM

You are right, it can be a 3D figure as well. Sorry for the confusion!

Hi generis pushpitkc ,

Could you please explain the logic behind highlighted texts?

Thanking you.

No problem, Do you think, maybe the question is incorrect?


Thanks,
GyM

PKN

I have assumed that the rectangular sandbox is actually a rectangle. Those are the formulae
for a rectangle's perimeter and area. I'm sure generis has assumed the same thing.

Hope this clears the confusion

Thank you pushpitkc

I have two points :-
1. Can we assume in "must be true question" ?
2. How can a rectangular box(=cuboid) be assumed as a rectangle, I mean a 3D figure to a 2D figure though it is known that the faces of the rectangular sand box
are rectangular surfaces.
Or, Is it a trap that we have to assume the given box as a rectangle since perimeter is given and perimeter is calculated for only 2-D surfaces.

I think the question may be missing something or I may be missing concepts.

Pardon me for any trouble.

Thanking you

PKN , questions are not trouble.
I'm not sure quite what you mean by this question.

Can we assume... What? This question does not provide three options, for example, from which we must choose one, two, three, all, or none.

Must be true seems to mean "only one of these equations is correct." As far as I can tell, "must be" means "is."

I think the word "box" in "sandbox" might be a trap if we overthink.

I do not think the question is flawed, and I do not think you are missing concepts. In fact, I think you grasped the logic quite well.

Either we ignore volume because we see it is not included, or we consider that we are dealing with a 3-D figure, but we are being asked 2-D questions.

We do not have enough typical information for a 3-D figure. Not one of the terms in any of the equations is cubed -- that is, volume is not at issue.

The emphasis is on rectangle -- perimeter and area.

If the given information's limits do not stop doubt about a 3-D prism, assume that the question asks about one face: the top of the sandbox.

In plane geometry:
-- P, perimeter is a linear measurement
-- A, area is measured in square units
-- V, Volume is measured in cubic units

We have L and W in linear feet
We do not have height, H
We have area, A, in square feet
There is no mention of cubic feet

Further, we have a rectangle.

A cube with equal W, L, and H would give us a third measurement that a rectangle cannot.

From those facts, I think we can infer that we are being asked about the sandbox as if it were a fenced-in piece of grass, for example.

Alternatively, we can infer from what is given that we are being asked about only one of the surface areas of the sandbox.

We can ask two dimensional questions about three dimensional figures.

As mentioned, this question asks about the top "face" of the sandbox.

I worked backwards from what we were given.

L and W will yield Perimeter
L and W will yield Area

L does not appear in the answers.
W\(^2\) appears in all the answers.

Area requires square units. W\(^2\) will provide square units.

Hence: use perimeter and width to redefine length in terms of W and P. The latter appear in all of the equations.

Use area to get a squared term.

Essentially, do we we understand the relationship among width, length, perimeter, and area?

Do we see that we must:
-- redefine length in terms of W and P
-- use the relationship between A and P, and
-- form an equation?

That's it.

I hope that helps.


**Some people in specialized areas calculate what they call the perimeter of a 3-D figure. That usually means the total length of the edges. I'm not a specialist in those areas. Nor do I think GMAT expects any of us to be such specialists.

generis Perimeter of a cuboid or any multi dimensional figure can definitely be calculated. The question in its premise, clearly says "rectangular sandbox" & not "a face of a rectangular sandbox". Assuming a Cuboid figure to be 2 dimensional is a bit too much of a leap of assumption, in my opinion.

I think the question is flawed & missing additional information. I doubt the GMAT will ever throw a question which is ambiguous in its structure.

If i were to design this question, i would clearly provide the relationship between L:W:H or mention that the question concerns only one face of the rectangular sandbox. If this information is added, then its not a bad question at all.

Request the experts Bunuel mikemcgarry chetan2u to evaluate the quality of this question.

Thanks,
GyM

Hi..

Your logic is absolutely correct.
There is nothing known as assumption here. It is clearly mentioned as a rectangular sandbox and it is a box whatever wildly you assume.
Perimeter and area have the same units of measurement and that too doesn't help here.

So yes the question is flawed.
It could have said..
# base of rectangular sandbox
# rectangular sheet

And you will be not left to imagine on your own in an official question.
NOTE- may be the word BASE has been missed out, a scenario that makes this question ok..

Phew!!, That puts me at ease now. Thank you Sire!!

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.