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# Is the area of the top of a rectangular pool larger than 1000 square

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Math Expert
Joined: 02 Sep 2009
Posts: 44586
Is the area of the top of a rectangular pool larger than 1000 square [#permalink]

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23 Jul 2015, 03:10
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Is the area of the top of a rectangular pool larger than 1000 square feet?

(1) The pool's top measures 50 feet diagonally.
(2) One side of the pool's top measures 25 feet.

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: Is the area of the top of a rectangular pool larger than 1000 square [#permalink]

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23 Jul 2015, 03:21
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Bunuel wrote:
Is the area of the top of a rectangular pool larger than 1000 square feet?

(1) The pool's top measures 50 feet diagonally.
(2) One side of the pool's top measures 25 feet.

Kudos for a correct solution.

Let, Length of Pool = L
Width of Pool = W
i.e. ARea of Pool = L*B

Question : Is L*B > 1000?

Statement 1: The pool's top measures 50 feet diagonally.
Diagonal = 50
Case-1: L=48, B=14, Area = 672 i.e. Less than 1000
Case-2: L=$$25\sqrt{2}$$, B=$$25\sqrt{2}$$, $$Area_{Max} = 1250$$ i.e. Greater than 1000
CONCEPT: Area of a rectangle for a given Diagonal will be Maximum when the sides of Rectangle are taken equal
NOT SUFFICIENT

Statement 2: One side of the pool's top measures 25 feet.
Second side can be very long or very short. Hence
NOT SUFFICIENT

Combining the two statements:
L^2 + B^2 = 50^2 and L=25
i.e. B can be calculated and the area will be exactly calculated which can definitely be compared with 1000. Hence,
SUFFICIENT

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Re: Is the area of the top of a rectangular pool larger than 1000 square [#permalink]

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23 Jul 2015, 04:18
Bunuel wrote:
Is the area of the top of a rectangular pool larger than 1000 square feet?

(1) The pool's top measures 50 feet diagonally.
(2) One side of the pool's top measures 25 feet.

Kudos for a correct solution.

Is l*b > 1000 ft^2?

Per statement 1, diag = 50 feet ---> the sides of the rectangle can be 30, 40 feet resp. Thus area = 30*40 = 1200 ft^2 . "Yes"

But if , $$b=10$$ ---> $$l = 20\sqrt{6}$$ ---->$$l*b = 10*20\sqrt{6}$$ = $$200 \sqrt{6}$$ < 1000. Thus "No". This statement is not sufficient.

Per statement 2, b =25. Without l, we can not find the area. Thus this statement is not sufficient.

COmbining, we get diag = 50, b =25 ---> $$l = \sqrt{1875}$$.

You do not have to calculate the area but only need to know that you will get 1 unique value for the area that will either be a "yes" for >1000 or a definite "no" for >1000.

Either way, both statements are sufficient when combined and hence C is the correct answer.
Math Expert
Joined: 02 Sep 2009
Posts: 44586
Re: Is the area of the top of a rectangular pool larger than 1000 square [#permalink]

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26 Jul 2015, 12:36
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Expert's post
Bunuel wrote:
Is the area of the top of a rectangular pool larger than 1000 square feet?

(1) The pool's top measures 50 feet diagonally.
(2) One side of the pool's top measures 25 feet.

Kudos for a correct solution.

800score Official Solution:

Statement (1) tells us the diagonal measure, but we don't know the dimensions. The pool's top might be very long and narrow, measuring less than a foot on one side. In this case its area would be well under 1000 square feet. The pool could also be 30 by 40 feet, with a diagonal of 50 feet and an area of 1200 square feet. So we can't answer whether the area of the pool is greater than 1000 square feet.

Statement (2) tells us the length of one side, but we need the length and the width of the pool to determine its area.

Combining the two statements, we could use the Pythagorean theorem to determine the missing dimension of the pool. We could then determine whether the area is larger than 1000 square feet. Since this is a Data Sufficiency question, you should not waste time with the calculation.Since combining the statements gives us sufficiency, the correct answer is choice (C).

Note: If you want to determine the actual area of the pool, you can use the properties of triangles to quickly calculate the area. Since the length of one side of the pool is half the length of the diagonal, we know that the triangle formed by the two sides of the pool and the diagonal is a 30-60-90 triangle. The side that is 25 feet must be the short side, and the longer side will be √3 times the length of the short side, or 25√3 feet.The area is thus 25 × 25√3 = 625√3, which is larger than 1000 square feet.
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Re: Is the area of the top of a rectangular pool larger than 1000 square [#permalink]

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10 Apr 2018, 06:42

Because the pool is rectangular, the long side and short side can be in any number of different proportions, which will affect the area. Knowing the diagonal allows us to form a right triangle but we need the measurement of the second side in order to calculate the missing measurement of the rectangle. Once we have both length and width of the rectangle we can accurately calculate the area and determine if it meets the criteria
Re: Is the area of the top of a rectangular pool larger than 1000 square   [#permalink] 10 Apr 2018, 06:42
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