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.
That’s wrong.
Why did the cylinder start flirting with the sphere?
Because he found her radius quite attractive!
But to answer your question, let's find the ratio of the height (h) to the base radius (r) of the cylinder.
First, let's consider the relationship between the cylinder and the inscribed sphere. The diameter of the sphere is twice the radius, so it will also be the diameter of the cylinder.
Now, the largest lateral area of the cylinder will be when it perfectly fits inside the sphere. This means that the diameter of the cylinder will be equal to the diameter of the sphere.
So, the base radius of the cylinder (r) will be half of the sphere's diameter, which is equal to the sphere's radius.
Therefore, the ratio of the height (h) to the base radius (r) is equal to the ratio of the height to the sphere's radius.
In mathematical terms:
h/r = h/(2r) = 1/2
So, the ratio of the height to the base radius of the cylinder with the largest lateral area is 1:2.
To find the ratio of the height to the base radius of the cylinder with the largest lateral area inscribed in a sphere, we can make use of some geometry principles.
Let's assume that the radius of the sphere is "r" and the height of the cylinder is "h", while the base radius of the cylinder is "R".
First, let's consider the cross-section of the cylinder and the sphere. In the cross-section, the diameter of the sphere is equal to the diagonal of the rectangle formed by the cylinder's base and height. This diagonal can be found using the Pythagorean theorem.
The diagonal of the rectangle formed by the base and height of the cylinder is equal to the diameter of the sphere, which is 2r. The sides of the rectangle are R and h.
According to the Pythagorean theorem:
(R^2) + (h^2) = (2r)^2
R^2 + h^2 = 4r^2
Now, we need to find the lateral area of the cylinder, which is given by:
Lateral Area = 2πRh
To maximize the lateral area, we need to maximize the product of R and h. Since we know that R^2 + h^2 = 4r^2, we can rewrite the equation as:
(R^2)(h^2) = (4r^2)(h^2) - (h^2)(h^2)
(R^2)(h^2) = (4r^2)(h^2 - R^2)
Since R^2 + h^2 = 4r^2, we can substitute this equation into the previous equation:
(R^2)(h^2) = (4r^2)(4r^2 - R^2)
Now, we want to find the ratio of the height to the base radius:
Ratio = h / R
By dividing both sides of the equation by (R^2), we get:
h^2 = (4r^2)(4r^2 - R^2) / R^2
h^2 = 4r^2(4r^2/R^2 - 1)
Thus, the ratio of the height to the base radius of the cylinder having the largest lateral area is:
h / R = √[4r^2(4r^2/R^2 - 1)] / R
Simplifying further, we get:
h / R = 2√[(4r^2/R^2 - 1)]
This is the ratio of the height to the base radius of the cylinder with the largest lateral area inscribed in the given sphere.
To find the ratio of the height to the base radius of the cylinder having the largest lateral area, we need to understand the relationship between the sphere and the cylinder.
Let's consider the sphere first. The sphere is inscribed within a cylinder, meaning that the diameter of the sphere is equal to the height of the cylinder. Let's call the radius of the sphere "r".
Now, let's focus on the cylinder. We're given that the cylinder has the largest lateral area when inscribed in the sphere. The lateral area of a cylinder is given by the formula: L = 2πrh, where "L" is the lateral area, "r" is the base radius, and "h" is the height of the cylinder.
Since the cylinder is inscribed within the sphere, the diameter of the sphere (which is equal to the height of the cylinder) is equal to twice the radius of the cylinder. Therefore, the height of the cylinder is 2r.
Substituting this value into the formula for lateral area, we get: L = 2πr(2r) = 4πr^2.
Now, let's find the ratio of height to base radius. The height of the cylinder is 2r, and the base radius is r. So, the ratio is: (2r) / r = 2.
Therefore, the ratio of the height to the base radius of the cylinder with the largest lateral area inscribed within the sphere is 2.
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