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D is the distance from vertex F to verte...

D is the distance from vertex F to vertex G. The base is square, and the height is twice the width. What is the volume of the solid in terms of d ?

A

`12d^(3) sqrt(6)`

B

`10d^(3) sqrt(5)`

C

`6d^(3)sqrt(2)`

D

`(d^(3)sqrt(6))/(18)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let the length and width be x and the height be 2x. Solve for x in terms of d.

Use the formula for the distance between opposite vertices :
`d=sqrt(l^(2)+w^(2)+h^(2))`
`d=sqrt(x^(2)+x^(2)+(2x)^(2))=sqrt(6x^(2))=sqrt(6)x`
`x=(d)/(sqrt(6))`
So for the dimensions of the solid we have `l = w = (d)/(sqrt(6)), h = (2d)/(sqrt(6))`.
Volume `=l xx w xx h = ((d)/(sqrt(6)))((d)/(sqrt(6)))((2d)/(sqrt(6)))`
`=(2d^(3))/(6sqrt(6))`
`= (d^(3))/(3sqrt(6))`
`= (d^(3))/(3sqrt(6))xx(sqrt(6))/(sqrt(6))`
`= (d^(3)sqrt(6))/(18)`
You also could Pick Numbers on this problem. Choose a value for x and then find the volume and the value of d. Plug d into the answer choices and see which one matches.
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