To find the volume of a triangular prism we use the equation 

In our case our 

Therefore,

The longest diagonal of a cube transverses the interior of the figure:

This distance is defined by the super pythagorean theorem :

where , , and  are the length, width, and height. Because the figure is a cube, all three are the same measure, and each is the side of the cube. 

We can use the given surface area to find the length of the side:

We can use this value for the side to plug into the super pythagorean theorem

Which can be simplified to

Quadrant IV consists of the points with positive -coordinates and negative -coordinates. Therefore is the correct choice.

The line of symmetry of the parabola of the equation

is the vertical line

Substitute :

The line of symmetry is

That is, the line of the equation .

Remember that the basic form of the equation of a circle is:

This means that the center point  is defined by the two values subtracted in the squared terms.  We could rewrite our equation as:

Therefore, the center is 

For this question, we will need to do three things:

1. Determine the point in question.
2. Use trigonometry to find the area of the angle in question.
3. Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore: 

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of  to  is here: 

Therefore, the angle is:

To solve for the sector area, we merely need to use our standard geometry equation. Note that the radius of the circle, based on the equation, is .

This rounds to .

For this question, we will need to do three things:

1. Determine the point in question.
2. Use trigonometry to find the area of the angle in question.
3. Use the equation for finding a sector area to finalize our answer.

Let us first solve for the coordinate by substituting into our equation:

Our point is, therefore: 

Now, we need to calculate the angle formed between the origin and the point that we were given. We can do this using the inverse tangent function. The ratio of  to  is here: 

Therefore, the angle is:

To solve for the sector area, we merely need to use our standard geometry equation. Note that , based on the equation, is .

This rounds to .

Since you are looking for a point on the , your  value will be zero.  

The center of the circle is at the origin and radius is the distance from the center, so that means the point you are looking for must be  points away from .  

This can be two points on the  but since you are looking for a positive one, your answer must be .

The displacement between negative three and positive one is four.

This mean that after flipping the point, it must be symmetrical to its original location.   The new point must also be 4 units to the right of the line.

The new point would be located at:  

The answer is:  

The relation 

is a circle with center  and radius  .

In other words, it is a circle with center at the origin, translated right  units and up  units (the radius is irrelevant to the question).

or

is this circle translated right zero units and up 2 units. The upshot is that the circle moves along the -axis only, and therefore is symmetric with respect to the -axis, but not the -axis. Also, as a consequence, it is not symmetric with respect to the origin.