# The total area of a cube is expressed by A(x) = 24x^2+24x+6. What is the volume of this cube?

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To find the volume of the cube, we need to determine the length of its edges. The total area of a cube is equal to six times the area of one face. So, we can express the area of one face as ( A(x) / 6 ), where ( x ) represents the length of an edge.

Given that ( A(x) = 24x^2 + 24x + 6 ), we can divide by 6 to find the area of one face:

( A(x) / 6 = (24x^2 + 24x + 6) / 6 = 4x^2 + 4x + 1 )

Now, we know that the area of one face of the cube is ( 4x^2 + 4x + 1 ). To find the length of an edge, we can solve for ( x ) by taking the square root of the area of one face.

( 4x^2 + 4x + 1 = (2x + 1)^2 )

Taking the square root of both sides:

( \sqrt{4x^2 + 4x + 1} = 2x + 1 )

Now, we solve for ( x ):

( 2x + 1 = \sqrt{4x^2 + 4x + 1} )

( 2x = \sqrt{4x^2 + 4x + 1} - 1 )

( x = \frac{\sqrt{4x^2 + 4x + 1} - 1}{2} )

Now that we've found the length of an edge in terms of ( x ), we can find the volume of the cube using the formula for volume:

( V = x^3 )

Substituting the expression we found for ( x ):

( V = \left(\frac{\sqrt{4x^2 + 4x + 1} - 1}{2}\right)^3 )

This expression represents the volume of the cube in terms of ( x ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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