The function {x} is defined as the area of a square with diagonal of

Question

The function {x} is defined as the area of a square with diagonal of length x. If x > 0 and {x^2} = x^2, what is the value of x?

(A) 1
(B) √2
(C) √3
(D) 2
(E) 4

IanStewart · Answer

If we draw the diagonal of a square, we divide the square into two 45-45-90 triangles, where the diagonal is the hypotenuse. Since in a 45-45-90 triangle, the sides are in a 1 to 1 to √2 ratio, if the diagonal of a square is of length d, each edge of the square is of length d/√2, and squaring that, the area of the square is d^2/2.

So if our function here, which I'll write as f(z) instead of {z} because it's confusing to use set brackets for anything other than sets, tells us the area of the square with diagonal x, then

f(z) = z^2/2

The question tells us something about f(x^2) for some number x, and substituting "x^2" for "z" in the above we find

f(x^2) = (x^2)^2/2 = x^4/2

We're told this equals x^2, so

x^4/2 = x^2
x^4 = 2x^2

and if x is nonzero, x^2 = 2, and x = √2 (since x is positive).

Chitra657 · Answer

VeritasKarishma 
Could you please explain.
Im getting 2 as the answer

Posted from my mobile device

Showmeyaa · Answer

All that this question is asking is to find a value of x such that it's area and diagonal are same.
We know that area of a square =  s^2 
Area of a diagonal =  s\sqrt{2} 
 s^2 = s\sqrt{2} 
 s = \sqrt{2}  - (B), IMO!

Kinshook · Answer

Given: The function {x} is defined as the area of a square with diagonal of length x. 
Asked: If x > 0 and {x^2} = x^2, what is the value of x?

{x} = Area of a square with diagonal of length x = x^2/2 since  diagonal = side *\sqrt{2} 
{x^2} =  \frac{x^4}{2} = x^2

x^2 = 2

x = √2

IMO B

KarishmaB · Answer

Function {x} = Area of square with diagonal of length x.

What is the area of a square if its diagonal is x? We know that area of a square is side^2. Also, in a square, Side = Diagonal/sqrt2.

So area of a square with diagonal x  = (\frac{x}{\sqrt{2}})^2 = \frac{x^2}{2}

Hence, {x}  = \frac{x^2}{2}

Then, what is {x^2}? 
Replace x by x^2.
{x^2}  = (\frac{(x^2)^2}{2}) = \frac{x^4}{2} 
(Note that you do not have to square the entire function. You only have to replace x by x^2.

We are given that this is also equal to x^2.

\frac{x^4}{2} = x^2

x^2(x^2 - 2) = 0 
 x = \sqrt{2}

Answer (B)

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The function {x} is defined as the area of a square with diagonal of

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