# 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 solid.please explain

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### 2 Answers

The volume of remaining solid is obtained subtracting the volume of the cone from the volume of cylinder.

The voume of cylinder is:

V cyl. = `pi` *r^2*h

V cyl. = `pi` *14^2*36

V cyl. = 7056 `pi`

The voume of cone:

V cone =`pi` ` ` *r^2*h/3

V cone =`pi` *49*24/3

V cyl. = 392 `pi` cm^3

The volume of remaining solid is:

V = V cyl. - V cone

V = 6664` pi` cm^3

The total surface area of remaining solid is the TSA of cylinder + TSA of cone - 2*area of base of cone:

TSA of cyl. = 2`pi` r(r + h)

TSA of cyl. = 2`pi ` *14(14+ 36)

TSA of cyl. = 1400 `pi` cm^2

TSA of cone = `pi` r(r+s)

We'll determine s using Pythagorean theorem:

s^2 = h^2 + r^2

s^2 = 24^2 + 7^2

s = sqrt(576+49)

s = 25

We'll consider only the positive value, since a length of a side cannot be negative.

TSA of cone = 7`pi` (7+25)

TSA of cone = 224 `pi`

TSA of the remained solid is;

TSA of cyl. + TSA of cone = 1400 `pi` + 224 `pi`- 98 `pi` = 1526 `pi` cm^2

**The volume and TSA of remainde solid are: V = 6664 `pi` cm^3 and TSA = 1526 `pi` cm^2.**

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