Sprinklers are being installed to water a lawn. Each sprinkler waters

The maximum distance four sprinkler can cover is 8 X 8 ( length and width)

Rectangle of area 32 can be in any dimension 1*32, 2*16, 8*4
For the rectangle of dimension 8*4 it can cover but for 2*16 it can't cover. since the maximum distance 4 sprinkler can cover is 8 in any direction

Posted from my mobile device

If we want to water the lawn completely, then any point on the lawn should be reached by at least one sprinkler.

1) We know that the lawn is rectangular with area 32, but no information is given about the radii of the circles that can be watered by the four sprinklers. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

2) We know the radii of the circles that can be watered by the four sprinklers, but no information is given about the dimensions of the lawn to be watered. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

1&2) If the rectangular lawn has length 64 and width 0.5, then it is too long to be watered completely by the four sprinklers because each is capable of watering only a circular area with diameter 4.

Let's try to find a different answer. What's the greatest rectangular area that one such sprinkler can completely water on its own? In any circle, the square is the inscribed rectangle with the greatest possible area. Since the diagonal of an inscribed square is also a diameter of the circle, the area in question is \(4^2/2=8\).

Four of these identical square areas can make a big rectangular area \(4\cdot 8=32\), so there's a case in which the lawn can be watered completely by the four sprinklers.

Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: E

Honestly I don't understand why E instead C.

St 1 and St 2 together is not sufficient to answer this query.

We need dimensions of the garden rather than area as a whole.

If dimensions are 1*32, ans is no, whereas If dimensions are 6*5.333, ans is yes, as diameter of each circle is 4 yards.

Hence, Ans E

sathik63
It can actually cover a rectangle with dimensions 16x2. Place the 4 circles side by side. Each has a dia 4.
It cannot cover 32x1 though.

This may help

Is this one of those hard questions from the new question set released recently by GMAC?

Yes. I've been working at the new book

PFA the OFFICIAL EXPLANATION

I'd like to disagree with the highlighted in red.
PFA the pics where the red regions are not covered.

I chose E, seeing that the condition B says "maximum of 2 yards" i.e. it is not necessarily 2 yards, hence condition B is not sufficient.

Is my understanding incorrect?

Cuz we dont know how the sprinklers are organized.

If they adjacent to each other too closely, they can’t water the entire lawn.

So to know that they can water the entire lawn or not, infos about organization of the sprinkler should be offered.

Posted from my mobile device

Hi there!

Each statement alone is clearly insufficient.

We will show both statements together are still insufficient, even in the particular case when the rectangle considered is a square.

It is easy to find a geometric configuration in which the answer to the question asked is "NO". Take the centers of all 4 circles at the center of the square, for instance.
(If you think the centers could not coincide, consider each one distinct to each other, but all of them very close to the center of the square.)

The geometric configuration shown in the last figure answers the question in the affirmative. To prove this is a viable possibility, please take into account two things:

1. In the figure on the left we present 4 "natural squares" that cover the big one.
2. In the central figure, we show that in the lengths offered, each of the 4 circles cover the corresponding "natural square".

The algebraic-geometric bifurcation presented guarantees the correct answer is (E).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Hi again.

This post was created in honor of careful readers.

In my last figure (above), all circles should pass through the center of the square.

The figure shown here is a better representation of what is really going on in the "YES" scenario.

Regards,
Fabio.

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Although the answer is E, I would like to point out the mistake some guys posted above. I am 100% confident to say that a 4 x 8 rectangle CANNOT be watered by these four sprinklers.

--->putting the 4 circles in collinear will not work, because there will be some area between the edge of the circles to the edge of the rectangle.
--->nor will putting 2 above and 2 below work. there will be room leftover between the edges not watered.

only a square with area of 32 works, and the 4 sprinkers must be placed in one defined way.

Statements combined:

Case 1: length = 32000 yards, width = 0.001 yard
In this case, the four sprinklers will clearly be unable to water all of the lawn, so the answer to the question stem is NO.

Case 2:


In the figure above:
Each circle represents the area watered by each sprinkler, while the red square represents the area of the rectangular lawn.
Since the radius of each circle = 2 yards, each blue line is composed of four 2-yard radii and thus has a length of 8.
Since the red square has a diagonal of 8, each side of the lawn = \(\frac{8}{\sqrt{2}}\)
Thus, the area of the lawn = \(\frac{8}{\sqrt{2}}*\frac{8}{\sqrt{2}} = 32\)
In this case, the four sprinklers can water all of the 32-square-yard lawn, so the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, the two statements combined are INSUFFICIENT.


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