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!!!

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺

teleportation

step-by-step explanation: