From solid cylinder of h= 36cm and r=14cm, a conical cavity of radius 7cm and height 24cm is drilled out. Find the volume and TSA of remaining...

the volume is simply the volume of the cone subtracted
from the volume of the cylinder

volume of cone = [pi *
(radius)^2 * height]/3

radius = 7, height =
24

volume of cone = [pi * (7)^2 *
24]/3

                      = [pi * 49 *
24]/3

                      = [1176 *
pi]/3

                      = 392 *
pi

volume of cylinder = pi * (radius)^2 *
height

radius = 14, height
36

volume of cylinder = pi * (14)^2 *
36

                          = pi * 196 *
36

                          = 7056 *
pi

volume of solid = 7056*pi -
392*pi

                      = 6664*pi
cm^3

total surface area is a bit
tricky

its not just the 2 surface areas added
together

you'll need the total surface area of the cylinder
but when the cone is drilled out you gain the surface area of the side of the cone but
you lose the base area.

surface area of cylinder =
2[pi*(radius)^2]+2*pi*radius*height

radius = 14, height =
36

surface area of cylinder =
2[pi*(14)^2]+2*pi*14*36

                                  =
2*pi*196+2*pi*504

                                 
=392*pi+1008*pi

                                 
=1400*pi

slantheight is the distance from the edge of the
base of a cone to the tip of the cone's point.  this is not given to us directly so we
must find it using pythagorean's theorem.

(slantheight)^2 =
(radius)^2 + (height)^2

slantheight =
sqrt[(radius)^2+(height^2)]

radius = 7, height =
24

slantheight =
sqrt[(7)^2+(24)^2]

                =
sqrt(49+576)

                =
sqrt(625)

                =
25

surface area of cone = pi * radius(radius +
slantheight)

radius = 7, slantheight =
25

surface area of cone =
pi*7(7+25)

                              =
pi*7*32

                              =
224*pi

so the total surface area of the solid is both
surface areas together but the base of the cone subtracted twice, once for the cone
itself and another from the cylinder.

base area of cone =
pi * radius^2

                          = pi *
7^2

                          =
49*pi

total surface area of solid = surface area of cone +
surface area of cylinder - 2*base area of cone

total
surface area of solid =
224*pi+1400*pi-2(49*pi)

                                     =
224*pi+1400*pi-98*pi

                                     =
1526*pi cm^2

volume of the remaining solid = 6664*pi
cm^3

total surface area of remaining solid = 1526*pi
cm^2