# math

A right circular cylinder is to be inscribed in a sphere of given radius. Find the ratio of the height to the base radius of the cylinder having the largest lateral area.

1. 👍 1
2. 👎 0
3. 👁 313
1. In my diagram , I arbitrarily let the radius of the cylinder be 1 unit
let the radius of the cylinder be r , and let the height of the cylinder be 2h
That way I can say:
r^2 + h^2 = 1
r^2 = 1 - h^2 or r = (1- h^2)^(1/2)

Surface area (SA) = 2πr^2 + 2πrh
= 2π(1-h^2) + 2π(1-h^2)^(1/2) h
= 2π(1-h^2) + 2π (h^2 - h^4)^(1/2)

d(SA)/dh = 2π [-2h + (1/2)(h^2 - h^4)((-1/2) (2h - 4h^3) ]
= 0 for a max of SA

(1/2)(2h-4h^3)/√(h^2 - h^4) = 2h
(1/2)(2h)(1 - 2h^2) / (h√(1-h^2) ) = 2h

(1 - 2h^2)/√(1 - h^2) = 2h
1 - 2h^2 = 2h√(1-h^2)
square both sides
1 - 4h^2 + 4h^4 = 4h^2 - 4h^4

8h^4 - 8h^2 + 1 = 0
solving this I got
h^2 = (2 ± √2)/4

case1: h^2 = (2+√2)/4
h = ..9238796 , r = ..38268.. , h/r = 2.41421.. or 1 + √2

case2: h^2 = (2-√2)/4
h = .38268... , r = .9238.. , h/r = .4142 .. or -1 + √2

Looking at Wolfram
http://www.wolframalpha.com/input/?i=maximize+2π%281-h%5E2%29+%2B+2π+%28h%5E2+-+h%5E4%29%5E%281%2F2%29

I will take case 2 as my answer.

Nasty, nasty question

1. 👍 0
2. 👎 1

## Similar Questions

1. ### calculus

A potter forms a piece of clay into a right circular cylinder. As she rolls it, the height h of the cylinder increases and the radius r decreases. Assume that no clay is lost in the process. Suppose the height of the cylinder is

2. ### Physics

Four objects - a hoop, a solid cylinder, a solid sphere, and a thin, spherical shell - each has a mass of 4.59 kg and a radius of 0.252 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in the

3. ### Calculus

Find the moment of inertia of a right circular cylinder of radius of base R and height H, about the axis of the cylinder, if the density at the point P is proportional to the distance from P to the axis of the cylinder. Write down

4. ### Calculus

A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 2 . A) What are the dimensions of such a cylinder which has maximum volume? B) What is the radius? C) What is the height?

1. ### Algebra

Hi! Can someone help me get started with these 6 problems, please? Would be a huge help! :D 1. A biologist studied the populations of common guppies and Endler’s guppies over a 6-year period. The biologist modeled the

2. ### math

A sphere of radius 3 is inscribed in a cylinder. What is the volume inside the cylinder but not inside the sphere?

3. ### Maths

Find The altitude of a right circular cone of maximum curved surface which can be inscribed in a sphere of radius 'r'.

4. ### Math

1. The radius of a circular disc is 5.8 inches. Find the circumference. Use pi = 3.14 18.212 in 36.424 in *** 42.34 in 54.63 in 2. Name the geometric solid shape suggested by a frozen juice can. Sphere Rectangular prism Pyramid

1. ### maths --plse help me..

Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone

2. ### Geometry

15.) Find the volume of a barber's pole having the shape of a right circular cylinder of radius 5 in. and height 29 in. topped by a sphere of the same radius. Round to the nearest tenth, if necessary. A) 2276.5 in.3 B) 2799.8 in.3

3. ### Calculus

A right circular cone is inscribed in a sphere of radius r. Find the dimensions of the cone that maximize the volume of the cone.

4. ### Math

Find the volume of the largest right circular cylinder (in units^3) that fits in a sphere of radius 4 units.