A cone of volume 54π is cut by a plane parallel to the base, 1/3 of the way up the height of the cone (from the base). Find the volume of the resulting frustum.

Correct  Vf = 38 π

Step-by-step explanation:

Given:

Vc = 54 π

The ratio of the height of the cone and the height of the frustum is:

Hc : Hf = 3 : 1  ⇒ Hc = 3 · Hf

The ratio of the radius R of the base of the cone to the radius r of the upper base of the frustum is:

R : r = Hc : (Hc - Hf)  ⇒ R : r = (3· Hf) : (3 · Hf - Hf) ⇒ R : r = 3 · Hf : 2 · Hf

R : r = 3 : 2 ⇒  r = (2/3) · R

The formula for calculating the volume of a cone is:

Vc = (R² · π · Hc) / 3 = (R² · π · 3 · Hf) / 3 = R² · π · Hf

The formula for calculating the volume of a frustum is:

Vf = (π · Hf · (R² + R · r + r²)) / 3 = (π · Hf · (R² + R · (2/3) · R + ((2/3) R)²)) / 3

Vf = (π · Hf · (R² + (2/3) ·R² + (4/9) · R²)) / 3 ⇒

Vf = (π · Hf · ( (9/9)R² + (6/9) ·R² + (4/9) · R²)) / 3 ⇒

Vf = (π · Hf · (19/9) · R²) / 3 = (19/27) · R² · π · Hf

Vc / Vf = (R² · π · Hf) / ((19/27) · R² · π · Hf)

R², π and Hf were shortened and we get:

Vc / Vf = 27 / 19  ⇒  Vf = (19 · Vc) / 27 ⇒

Vf = 19 · 54 · π / 27 = 38 π

Vf = 38 π

God is with you!!!

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕

42.4

step-by-step explanation:

convert to decimal