What is the maximum volume of a cylinder inscribed in a sphere?
A sphere of fixed radius (R) is given. Let r and h be the radius and the height of the cylinder respectively. Hence, the volume of the cylinder is the maximum when the height of the cylinder is 2 R / √ 3.
What is the maximum volume of a cylinder?
Therefore the volume is a maximum when 2r−2h+h=0, so h=2r and hr=2. Show activity on this post. Show activity on this post. Let the ratio of height to radius be ρ, then h=ρr.
How many sphere can fill a cylinder?
All Answers (5) One can from there estimate how many such spheres will fit. Your cylinder has volume V_c=pi*D^3. Your spheres are each V_s= 4/3*pi*d^3.
How many spheres can fit in a sphere?
Sphere packing in a sphere
| Number of inner spheres | Maximum radius of inner spheres | Packing density |
|---|---|---|
| Approximate | ||
| 1 | 1.0000 | 1 |
| 2 | 0.5000 | 0.25 |
| 3 | 0.4641… | 0.29988… |
What is the volume of the sphere to the cylinder?
Today, we are blessed with the understanding Calculus, and formulas for the volumes of basic shapes are taught to all in High School. What pleased Archimedes so much about the solution is that the ratio of the volume of the sphere to the cylinder is the same as the ratio of the surface area of the sphere to the surface area of the cylinder!
How to find the volume of a cylinder using the Pythagorean theorem?
By the Pythagorean theorem, the radius of the cylinder is given by r2 = R2 − (h 2)2. The volume of the cylinder is hence V = πr2h = π (hR2 − h3 4). Differentiating with respect to h and equating to 0 to find extrema gives dV dh = π (R2 − 3h2 4) = 0 ∴ h0 = 2R √3.
What is the volume of a hollow cylinder?
Tadaaam! The volume of a hollow cylinder is equal to 742.2 cm 3. Remember that the result is the volume of the paper and the cardboard. If you want to calculate how much plasticine you can put inside the cardboard roll, use the standard formula for the volume of a cylinder – the calculator will calculate it in the blink of an eye!
How do you find the volume of a slanted cylinder?
Find the radius, side length and slant angle of the cylinder. Square the radius. Multiply the result by pi. Take the sin of the angle. Multiply the sin by the side length. Multiply the result from step 3 and 5 together. The result is the slanted volume.