A figure is located at (2, 0), (2, −2), (6, −2), and (6, 0) on a coordinate plane. What kind of 3-D shape would be created if the figure was rotated around the x-axis? Provide an explanation and proof of your answer to receive full credit. Include the dimensions of the 3-D shape in your explanation.

Step-by-step explanation:

First we need to make a little representation to get an idea on what's going on

After doing this we notice that the shape we get a rectangle.But noticing isn't enough we should prove it mathematically :

how ?

let's start first by naming our rectangle :  A(2.0)  B(6.0) C(2.-2) D(6.-2)

This will make the task easier.

Now time to use vectors :

[AB] ⊥ [BD] ⇒ VECTOR AB .VECTOR BC = NULL VECTOR

VECTOR AB (6-2,0-0) VECTOR BC (6-6,-2-0)

Now V(AB) . V(BC) = (6-2)*(6-6)+(0-0)(-2-0)

                                  = 0-0

                                   = NULL VECTOR

[AD] ⊥ [DC]  ⇒ VECTOR AD .VECTOR DC = NULL VECTOR

VECTOR AD (2-2,-2-0)VECTOR  DC (6-2,-2+2)

Now V(AD) . V(DC) = (2-2)*(6-2)+(-2+2)*(2-0)

                                 =  0-0

                                 = NULL VECTOR

Now all we should do is to prove that at least 2 edges are equal to each other .[AB] and [DC] AB= =4DC= = 4As predicted AB=DC When we rotate the trigle around the x-axis, we are rotating it around AB wich means that the 3D shape we well get is a cylinder

Let r be the radius of the cylinder's base and h its height.

we deduce that h = AB = DC so h= 4The diameter of the base equal to AB wich mean that r=AB/2=2 The trick is to represent the information and then start asking questions : How can i prove this ?