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The diagonal AC of the rectangular solid forms a 60° angle with the di

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Bunuel
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The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 01:15
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The diagonal AC of the rectangular solid forms a 60° angle with the diagonal of its base. Given the dimensions in the figure, what is the volume of the rectangular solid?


A. \(40\sqrt{3}\)

B. \(50\sqrt{3}\)

C. \(60\sqrt{3}\)

D. \(70\sqrt{3}\)

E. \(80\sqrt{3}\)


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C
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 01:45
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Given two sides and 90-degree right angle property.
AB length = 5 (3-4-5 right angle triangle)
Now,
Triangle ABC is a 30-60-90 = X:X√3:2X right-angle triangle whose shortest side is 5(opposite to 30)
using the property,
the longest side (opposite to 90), AC will be 10 (2*5= 10)
Also, AC is the longest side of cuboid.
AC = √l^2+b^2+h^2
10 = √3^2+4^2+h^2
100 = 9+16+h^2
75 = h^2
h = 5√3

Volume of cuboid = l*b*h
=> 3*4*5√3
=> 60√3

Answer is C
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 04:45
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AB = 5, which is the side opposite 30°.

In a 30 - 60 - 90 triangle, the ratio of sides opposite 30 - 60 - 90 = 1 : √3 : 2

Therefore \(\frac{5}{CB} = \frac{1}{{\sqrt{3}}\).

CB = 5√3

The volume = l * b * h = 3 * 4 * 5√3 = 60√3


Option C

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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 05:03
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Answer C
Main diagonal of the cuboid = 5/sin(60 deg) = 10
Height of the cuboid= 10(cos30 deg) = 5 \sqrt{3}
Volume= 4 x 3 x 5 \sqrt{3} = 60 \sqrt{3}



Bunuel wrote:

The diagonal AC of the rectangular solid forms a 60° angle with the diagonal of its base. Given the dimensions in the figure, what is the volume of the rectangular solid?


A. \(40\sqrt{3}\)

B. \(50\sqrt{3}\)

C. \(60\sqrt{3}\)

D. \(70\sqrt{3}\)

E. \(80\sqrt{3}\)


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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 05:06
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Bunuel wrote:

The diagonal AC of the rectangular solid forms a 60° angle with the diagonal of its base. Given the dimensions in the figure, what is the volume of the rectangular solid?


A. \(40\sqrt{3}\)

B. \(50\sqrt{3}\)

C. \(60\sqrt{3}\)

D. \(70\sqrt{3}\)

E. \(80\sqrt{3}\)


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pythagora: a^2+b^c=diagonal^2=9+16=25, diag=5
specials: 30-60-90=x-xV3-2x=5-5V3-10
volume: abc=3*4*c=12c=12(5V3)=60V3
(C)
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 13:10
calculated it to be 60 root 3. Official answer is missing from this post. can somebody provide that
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The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   24 Nov 2020, 20:31
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Bunuel wrote:

The diagonal AC of the rectangular solid forms a 60° angle with the diagonal of its base. Given the dimensions in the figure, what is the volume of the rectangular solid?


A. \(40\sqrt{3}\)

B. \(50\sqrt{3}\)

C. \(60\sqrt{3}\)

D. \(70\sqrt{3}\)

E. \(80\sqrt{3}\)


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1. Use pythagoras theorem property to calculate the value of AC.
This will be 5.

2. Now, we know that sides in a 30-60-90 triangle are in the ratio of 1:\sqrt{3}:2
Hence, sides will be 10:5\sqrt{3}:5

This will give us the height.

Alternatively, the longest diagonal in a rectangle is given by : \sqrt{l^2 + b^2 + h^2}
So, volume = lbh
=3*4*5√3
=60√3

IMO C
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   30 Nov 2020, 11:38
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Bunuel wrote:

The diagonal AC of the rectangular solid forms a 60° angle with the diagonal of its base. Given the dimensions in the figure, what is the volume of the rectangular solid?


A. \(40\sqrt{3}\)

B. \(50\sqrt{3}\)

C. \(60\sqrt{3}\)

D. \(70\sqrt{3}\)

E. \(80\sqrt{3}\)



Solution:

We are given the length and width of the rectangular solid. If we can determine the height, then we can determine the volume of the solid. Notice that AB = 5 since it’s the hypotenuse of a 3-4-5 right triangle. Now, since triangle ABC is a 30-60-90 right triangle and the shorter leg, AB, is 5, the longer leg, BC, is 5√3. Since BC is also the height of the solid, we see that the height of the solid is 5√3, and thus its volume is 3 x 4 x 5√3 = 60√3.

Answer: C
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink] New post   30 Nov 2020, 12:58
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Given

    • A rectangular solid given in the figure.
    • The rectangular solid forms a 60° angle with the diagonal of its base.
    • Two dimensions are given in the figure as 3 and 4.

To Find

    • The volume of the rectangular solid.


Approach and Working Out


    • Using Pythagorean triplet we can say, AB = 5 (The triplet used is 3: 4: 5)

    • The triangle ABC is 30-60-90 triangle.
      o AB = 5 (Opposite to 30 degree)
      o BC = 5√3

    • Volume = 3 × 4 × 5√3 = 60√3

Correct Answer: Option C
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Re: The diagonal AC of the rectangular solid forms a 60° angle with the di [#permalink]
30 Nov 2020, 12:58
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