Select the figure obtained when rotating the figure about an axis along its largest side, and find the exact surface area of the resulting figure.

a rectangle with length 22 and width 18.

the figure obtained is a (square/cylinder/sphere)

the exact surface area is π.

The figure obtained is a cylinder

The exact surface area is 1440π.

Step-by-step explanation:

* Lets study the rotation of a rectangle around one of its sides

- If the dimensions of the rectangle are length (L) and width (W)

- If we rotate the rectangle around its length (L), we will construct

 a cylinder with radius = W and height = L

- If we rotate the rectangle around its width (w), we will construct

 a cylinder with radius = L and height = W

∵ The surface area of any cylinder =

   perimeter of base × height + 2 base area

∵ The base is a circle

∴ perimeter base = 2πr and base area = πr²

∴ The surface area = 2πrh + 2πr²

* In first case:

- Surface area = 2π(W)L + 2π(w)² = 2πWL + 2πW²

* In second case:

- Surface area = 2π(L)W + 2π(L)² = 2πLW + 2πL²

* Now lats check our question:

∵ L = 22 and W = 18

- It will rotate around the largest side

∴ It will rotate around L

∴ The figure obtained is a cylinder

* From the explanation above this is the first case

∵ L = 22 and W = 18

∵ The surface area = 2πWL + 2πW²

∴ The surface area = 2 × 18 × 22 × π + 2 × (18)² × π

                                = 792π + 648π = 1440π

* The exact surface area is 1440π.

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.