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

Question

Maverick Smith · Accepted Answer

#8x^3+12x^2+6x+1#

I'm going to assume you meant the surface area is given by #A(x)#.

We have #A(x)=24x^2+24x+6#

The formula for the surface area of a cube is given by #6k^2#, where #k# is the length of a side.

We can say that:

#6k^2=24x^2+24x+6#

#k^2=4x^2+4x+1#

#k^2=(2x+1)^2#

#k=2x+1#

So the length of a side is #2x+1#.

On the other hand, #V(x)#, the volume of he cube, is given by #k^3#.

Here, #k=2x+1#

So we can say:

#V(x)=k^3=(2x+1)^3#

#V(x)=(2x+1)^2(2x+1)#

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

#V(x)=8x^3+12x^2+6x+1#

So the volume of this cube is given by #8x^3+12x^2+6x+1#

Noah Atkinson · Answer

undefined 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 $.