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655-705 (Hard)|   Geometry|      
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Bunuel
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A right circular cylinder has a height of 20 and a radius of 5. A rect 29 Dec 2015, 07:57
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55% (hard)

Question Stats:

69% (02:54) correct 31% (02:54) wrong based on 217 sessions

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A right circular cylinder has a height of 20 and a radius of 5. A rectangular solid with a height of 15 and a square base, is placed in the cylinder such that each of the corners of the solid is tangent to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. What is the volume of the liquid?

A. 500(π – 3)
B. 500(π – 2.5)
C. 500(π – 2)
D. 500(π – 1.5)
E. 500(π – 1)
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D
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A right circular cylinder has a height of 20 and a radius of 5. A rect 29 Dec 2015, 09:07
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D
Volume of cylinder = pie*r^2*h = 500pie
Now, base of the inner solid is a square.
Since all the corners of the quadrilateral touch the circle, length of diagonals is the diameter which is 10.
Therefore area of the square = 1/2*(diagonal)^2 = 50
Now volume of any regular pyramid id (are of base)*height
Therefore, are of the inner solid is 50*15 = 750

The available volume is 500*pie - 750 = 500(pie - 1.5)
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A right circular cylinder has a height of 20 and a radius of 5. A rect 30 Dec 2015, 18:01
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[quote="Bunuel"]A right circular cylinder has a height of 20 and a radius of 5. A rectangular solid with a height of 15 and a square base, is placed in the cylinder such that each of the corners of the solid is tangent to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. What is the volume of the liquid?

The square base has sides of sqrt(50) due to the 45-45-90 triangle

20*25*pi - 15*sqrt(50)^2
=500(π – 1.5)


D. 500(π – 1.5)
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A right circular cylinder has a height of 20 and a radius of 5. A rect 26 Oct 2020, 00:21
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Start with the Bottom Face of the Rectangular Solid that is a Square. Since Each of the Corners of the Solid is Tangent to the Cylinder, the Bottom Face Square of the Rect. Solid will be INSCRIBED Inside the Circle on the Bottom of the Cylinder.


Rule: when a Square is Inscribed Inside a Circle, the Diameter of the Circle = Diagonal of the Square

Diagonal of a Square = (Side of Square = s) * sqrt(2)

Diameter of Circle on Bottom of Cylinder = 2 * 5 = 10

(s) * sqrt(2) = 10

(s) = (10) / (sqrt(2))


The Dimension of the Rectangular Solid are:

10 / sqrt(2)

10 / sqrt(2)

and

15


To Find the Volume of the Liquid that fills the Cylinder AROUND the Inserted Rect. Solid, we need to find:

(Volume of Cylinder) - (Volume of Rectangular Solid) =

[ (pi) * (r)^2 * H ] - (L * W * H) =

[ (pi) * (5)^2 * 20 ] - ( 10/sqrt(2) * 10/sqrt(2) * 15) =

500(pi) - ( 10 * 10 * 15 / (sqrt(2))^2 ) =

500(pi) - (1,500 / 2) =

500(pi) - 750 =

------taking 500 Common to match the Answer Choices-----

500(pi) - 750 = 500 * ( (pi) - 750/500) = 500 * ( (pi) * 3/2 ) =

500 * ( (pi) - 1.5)

-D-
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A right circular cylinder has a height of 20 and a radius of 5. A rect 07 Nov 2020, 12:29
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Fdambro294
Start with the Bottom Face of the Rectangular Solid that is a Square. Since Each of the Corners of the Solid is Tangent to the Cylinder, the Bottom Face Square of the Rect. Solid will be INSCRIBED Inside the Circle on the Bottom of the Cylinder.


Rule: when a Square is Inscribed Inside a Circle, the Diameter of the Circle = Diagonal of the Square

Diagonal of a Square = (Side of Square = s) * sqrt(2)

Diameter of Circle on Bottom of Cylinder = 2 * 5 = 10

(s) * sqrt(2) = 10

(s) = (10) / (sqrt(2))





The Dimension of the Rectangular Solid are:

10 / sqrt(2)

10 / sqrt(2)

and

15


To Find the Volume of the Liquid that fills the Cylinder AROUND the Inserted Rect. Solid, we need to find:

(Volume of Cylinder) - (Volume of Rectangular Solid) =

[ (pi) * (r)^2 * H ] - (L * W * H) =

[ (pi) * (5)^2 * 20 ] - ( 10/sqrt(2) * 10/sqrt(2) * 15) =

500(pi) - ( 10 * 10 * 15 / (sqrt(2))^2 ) =

500(pi) - (1,500 / 2) =

500(pi) - 750 =

------taking 500 Common to match the Answer Choices-----

500(pi) - 750 = 500 * ( (pi) - 750/500) = 500 * ( (pi) * 3/2 ) =

500 * ( (pi) - 1.5)

-D-
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can you explain this part? ------taking 500 Common to match the Answer Choices-----

500(pi) - 750 = 500 * ( (pi) - 750/500) = 500 * ( (pi) * 3/2 ) =

500 * ( (pi) - 1.5)
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A right circular cylinder has a height of 20 and a radius of 5. A rect 07 Nov 2020, 20:35
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Sure

Think of it like this. If you had to Multiply the 500 we factored out, what would you need to multiply it by to get back to the Original 750?

500 * (1.5) = 750

Thus, when you factor out the 500, think of it as leaving 1.5 in its place

Just like if you had:

2x + 6

2 * (x + 3)

After factoring out the 2, you had to leave 3 in its place. If you were to multiply 2 back through the parenthesis again, you would end up with:

2x + 6 again

Does that make sense?

In hindsight, I should have just left the 1.5 behind instead of 750/500, but I take my “steps” to the extreme sometimes lol

There are times when you find an answer and you need to make it match the answer choices. This was one of those times.

And even though the percentage seems high for a “hard” question, this was a very difficult question.



baraa900
Fdambro294
Start with the Bottom Face of the Rectangular Solid that is a Square. Since Each of the Corners of the Solid is Tangent to the Cylinder, the Bottom Face Square of the Rect. Solid will be INSCRIBED Inside the Circle on the Bottom of the Cylinder.


Rule: when a Square is Inscribed Inside a Circle, the Diameter of the Circle = Diagonal of the Square

Diagonal of a Square = (Side of Square = s) * sqrt(2)

Diameter of Circle on Bottom of Cylinder = 2 * 5 = 10

(s) * sqrt(2) = 10

(s) = (10) / (sqrt(2))





The Dimension of the Rectangular Solid are:

10 / sqrt(2)

10 / sqrt(2)

and

15


To Find the Volume of the Liquid that fills the Cylinder AROUND the Inserted Rect. Solid, we need to find:

(Volume of Cylinder) - (Volume of Rectangular Solid) =

[ (pi) * (r)^2 * H ] - (L * W * H) =

[ (pi) * (5)^2 * 20 ] - ( 10/sqrt(2) * 10/sqrt(2) * 15) =

500(pi) - ( 10 * 10 * 15 / (sqrt(2))^2 ) =

500(pi) - (1,500 / 2) =

500(pi) - 750 =

------taking 500 Common to match the Answer Choices-----

500(pi) - 750 = 500 * ( (pi) - 750/500) = 500 * ( (pi) * 3/2 ) =

500 * ( (pi) - 1.5)

-D-


can you explain this part? ------taking 500 Common to match the Answer Choices-----

500(pi) - 750 = 500 * ( (pi) - 750/500) = 500 * ( (pi) * 3/2 ) =

500 * ( (pi) - 1.5)
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A right circular cylinder has a height of 20 and a radius of 5. A rect 08 Apr 2021, 20:18
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Just wondering here if GMAT ever throws in unnecessary details. I can't say I've seen many questions that do include such details.

This one here...the volume is completely irrelevant. The question amounts to asking what's the difference b/w the area of the cylinder and box.
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A right circular cylinder has a height of 20 and a radius of 5. A rect 19 Aug 2023, 10:00
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A right circular cylinder has a height of 20 and a radius of 5. A rectangular solid with a height of 15 and a square base, is placed in the cylinder such that each of the corners of the solid is tangent to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. What is the volume of the liquid?

SOLUTION: The diagonal of the square base will be equal to the diameter of the cylinder in which it is placed = 10
diagonal of square = [(2)^0.5]a,
where a is the side of the square base
=>[(2)^0.5]a=10
=>a=10/[(2)^0.5]
=>Volume of rectangular solid = l*(square base side)^2
=>15*(a)^2 = 15(10/root2)^2 =15*(100/2)=750
volume of cylinder will be- pi [(r)^2] h = pi*25*20=500pi
the volume of liquid will be = volume of cylinder-volume of rectangular solid
= 500pi-750
= 500(pi-750/500)
= 500(pi-3/2) => 500(pi-1.5)

Hence answer D
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