Math Is Fun Forum

gourish

Member

Registered: 2013-05-28

Posts: 153

functions related to geometry

"A tower has the following shape: a right circular truncated right circular cone (one with radii 2R(the lower base) R(the upper base), and height R bears a right circular cylinder whose radius is R and height is 2R. Finally a semisphere of radius R is mounted on the cylinder. suppose the cross sectional area S of the tower is given by f(x) where x is the distance of the cross section from the lower base of the cone"
define the function.
i know that the function is (pie)*R^2 for the interval R≤ x≤ 3R
but what about the other intervals and yeah the domain of the function is is from 0 to 4R. i am facing problems with relating the area of the cross section with the radii and x of both the cone and semisphere help me please!

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: functions related to geometry

hi gourish,

Let's see if I can 'talk' my way up this tower.

(i) At the bottom it has radius 2R and this tapers to a radius of R over a distance of height R

(ii) Then it becomes a cylinder.  I agree with your formula for that.

(iii) At the top it is half a sphere, radius R. 

So now for the formulas:

I'll do the radius at each stage.  You can convert that to the area of a circle (they all have circular cross sections)

(i)  This is a linear function as the taper is regular.  I'll use y for the radius at any point

At x = 0, y = 2R.  At x = R, y = R.  So if you graphed this the gradient would be -1 (R across is R down * that's putting the tower on its side so that x is across as usual and y up.  I've added a diagram to show this)

y = 2R -x

(iii) Here the x and y coordinates lie on the sphere so you can use Pythagoras to write an equation linking them:

(x-3R)^2 + y^2 = R^2

Re-arrange to make y the subject:

y =  √(R^2 - [x-3R]^2)

Hopefully that should do it. 

Bob

ps.  Looks a bit like a Dalek. 

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: functions related to geometry

that was a great explanation thanks bob. it was really helpful but i have another of similar kinda this time i am asked to find out the volume of triangle inscribed in a sphere of radius R. where x is the height of the triangle inscribed in it. i am having troubling finding the domain of the function as a starter and that's the reason why i am not even able to define this function. hope you can explain it to me like you did for my earlier question. i would really appreciate it

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: functions related to geometry

volume of triangle inscribed in a sphere

triangle??

Do you mean cone?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: functions related to geometry

well i posted that question on the forum

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: functions related to geometry

What has me puzzled is you have 'triangle'.  That is a flat shape and has no volume. ???

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: functions related to geometry

it was my mistake the actual question was about a cone which i posted already and found the answer

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"The man was just too bored so he invented maths for fun"
-some wise guy