In the rectangular solid depicted above, AB = 6, BC = 8, CD = 5, and A

Question

https://gmatclub.com/forum/download/file.php?id=55513 In the rectangular solid depicted above, AB = 6, BC = 8, CD = 5, and AE > EB > 2. Which of the following could be possible values for the volume of the shaded area? I. 150 II. 170 III. 180 A. I only B. II only C. I and II only D. II and III only E. I, II, and III 1.jpg

yashikaaggarwal · Answer

Area of cuboid = 2(lb+BH+hl)
=> 2(6*8+8*5+5*6)
=> 2(48+40+30)
=> 2(118)
=> 236

The lowest possible area of shaded part is when the area of prism is greatest.

The base of prism = AE greatest value is 3.9 (3.9>2.1>2)
The area of prism = 1/2*height*area of new base of prism*breadth of box

=> 5*3.9*4
=> 78

The minimum area of the shaded part can be 236-78 = 158

Similarly,
The largest possible area of shaded part is when the area of prism is least.

The base of prism = AE least value is 3.1 (3.1>2.9>2)
The area of prism = 1/2*height*area of new base of prism*breadth of box

=> 5*3.1*4
=> 62

The maximum area of the shaded part can be 236-62 = 174

The value of shaded region lies in between 158 < S < 174

And only 2nd is satisfying the equation.

IMO B

Posted from my mobile device

Zuki1803 · Answer

the shaded region is a trapezium
so area = 1/2(sum of || sides)*h = (1/2(6+2.1)*8)
Area into width*volume=32.4*5= 162 as min
now AE>BE>2 can take values from 3.9>2.1>2 to 3.1>2.9>2
so area = 1/2(sum of || sides)*h = (1/2(6+2.9)*8)
Area into width*volume=35.6*5=178 as max
hence 162<v<178

Regor60 · Answer

The volume of the shaded region is the volume of the entire solid less the volume of the triangular solid. The volume of the triangular solid is half the volume of the rectangular solid formed by rotating the triangular solid around its diagonal: 1/2(AE*BC*CD) So the volume of the shaded region is: AB*BC*CD - 1/2(AE*BC*CD) = 240 - 1/2(40AE) = 240 - 20AE Given the constraints, AE>EB, means AE>3 and 22, this answer fails 170 = 240 -20AE or AE = 3 1/2 and EB = 2 1/2 This is a possible answer 180 = 240 - 20AE or AE = 3. This is not a possible answer So only II works Posted from my mobile device

samarpan.g28 · Answer

Some formulas which will help us:
1. Volume of the rectangular cube = length*width*height  unit^3  - here it is 6*8*5=240  cm^3 .
2. Volume of the unshaded triangular prism = area of the traingle*depth, where area of the triangle is 1/2 * base * height. Here the base is AE (let AE=x), height=8cm (same as BC) and depth is 5cm (same as CD). Therefore, the volume is 1/2 * x * 5 * 8 = 20x  cm^3 .

We need to find x. We pick each option and validate this condition given AE>EB>2. Also given, AE+EB=AB=6cm.

Option I: if the volume of shaded area is 150 then the unshaded i.e. triangular prism's volume is 240-150=90.
now 20x=90 or, x=4.5, so AE=4.5 and EB=6-4.5=1.5 and AE>EB but EB is not greater than 2. Wrong.
Option II: 240-170=70.
20x=70 or, x=3.5. AE=3.5 and EB=6-3.5=2.5 so AE>EB and EB>2. Correct.
Option III: 240-18-=60.
20x=60 or, x=3. AE=3 and EB =6-3 =3 so AE is not greater than EB, however, EB>2. Wrong.

Only II is correct. Option (B).

In the rectangular solid depicted above, AB = 6, BC = 8, CD = 5, and A

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In the rectangular solid depicted above, AB = 6, BC = 8, CD = 5, and A

Prep Toolkit

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