Calculating Cone Properties: Height, Slant, Radius, and Volume in Terms of x

[rachael]

1.A cone where h= height of the cone
s= slant of the cone
r= radius of the cone



a.Find c,circumference of a cone in terms of x
b.find r, radius of the cone in terms of x
c. find an expression for the slant height of the cone in terms of x
d. find h, hieght of the cone in terms of x
e. find the volume of the curve in terms of x in the form of
V= (a-x)^2[(bx-x^2)^(1/2)]
-----------------------
c where a,b,c are positive
f find the domain of the function
g. find v'(x)

what i have done:
a. x=2pir
b. r=x
---
2pi
c. x=s
d. i don't know
e. i don't know
f. [0,2pi]
g.?

could someone please help me
thank you

 

[Dr.Brain]

its easy actually...draw a cone ...and then draw an imaginary circle along the contour of the cone ...and label its radius as r , and let it be at height of x from below... ...therefore height of the cone above this cricle will be [h-x] , so u get a smaller cone (upper one)... use pythagorus theorem , u can get the slant height of upper smaller cone using property of similar triangles...

 

[HallsofIvy]

Frankly, the problem doesn't make a whole lot of sense. You are given h and r but then asked to write things "in terms of x"?? What is x? You appear to be assuming that x is r, even when the problem says "find r in terms of x", but that is not given in your problem.

Assuming that x really is just r, then as Dr. Brain says- draw a picture. Looking at the cone from the side, it is an isosceles triangle height h and base 2r so each half is a right triangle with height h and base r. The slant height is the length of the hypotenuse so you can use the Pythagorean theorem.

But then I run into "d. find h, hieght of the cone in terms of x".
You are given two independent values, h and r. If x really is r, then there is no way to right h "as a function of x". Knowing r tells you nothing about h.

"e. find the volume of the curve in terms of x in the form of
V= (a-x)^2[(bx-x^2)^(1/2)]"

The volume of the curve? Do you mean volume of the cone? The volume of a cone depends on both the height, h, and the radius, r, and they are independent. The volume cannot be written as a function of a single variable unless there is some relation between r and h you haven't given here.

 

[rachael]

Frankly, the problem doesn't make a whole lot of sense. You are given h and r but then asked to write things "in terms of x"?? What is x? You appear to be assuming that x is r, even when the problem says "find r in terms of x", but that is not given in your problem.

"h" and "r" is just a label for the cone. Assuming the "h" and "r" is not there. Therefore we need to look for c and r in terms of x.




"e. find the volume of the curve in terms of x in the form of
V= (a-x)^2[(bx-x^2)^(1/2)]"

The volume of the curve? Do you mean volume of the cone? The volume of a cone depends on both the height, h, and the radius, r, and they are independent. The volume cannot be written as a function of a single variable unless there is some relation between r and h you haven't given here.

oops i meant cone

 

[HallsofIvy]

Did you understand anything I wrote? The problem, as you stated it, still doesn't make sense. If you were really told to express these things "in terms of x", what in the world is x??