Find the diameter and the height of a cylinder of maximum volume which can cut from a sphere of radius 12cm
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To find the maximum volume of a cylinder that can be cut from a sphere, we need to maximize the volume function for a cylinder.
Let's start by finding the volume formula for a cylinder:
V = πr²h
Where V is the volume, r is the radius of the cylinder's base, and h is the height of the cylinder.
Given that a sphere has a radius of 12 cm, the diameter of the sphere is 2 * 12 cm = 24 cm.
In order to maximize the volume of the cylinder, the diameter should be equal to the height.
So, let's substitute r = 12 cm and h = 12 cm in the volume formula:
V = π(12)²(12) = 1728π cm³
Therefore, the diameter and height of the cylinder with maximum volume that can be cut from the sphere are both 12 cm.
To find the maximum volume of a cylinder that can be cut from a sphere, we need to understand the relationship between the sphere and the cylinder.
First, let's consider the properties of a sphere. The volume of a sphere is given by the formula:
V_s = (4/3)πr^3
where V_s is the volume of the sphere and r is the radius.
Now, let's consider a cylinder inscribed within the sphere. The diameter of the cylinder will be equal to the diameter of the sphere, and the height of the cylinder will be equal to the diameter of the sphere.
To maximize the volume of the cylinder, we can use the following steps:
Step 1: Find the radius of the sphere.
Given the sphere has a radius of 12 cm, the radius of the sphere is 12 cm.
Step 2: Find the diameter and height of the cylinder.
Since the diameter of the cylinder is equal to the diameter of the sphere, it is also equal to 2 times the radius of the sphere. Therefore, the diameter of the cylinder is 2 * 12 cm = 24 cm.
Similarly, the height of the cylinder is equal to the diameter of the sphere, which is also equal to 2 times the radius of the sphere. Therefore, the height of the cylinder is 2 * 12 cm = 24 cm.
So, the diameter of the cylinder is 24 cm, and the height of the cylinder is 24 cm.
Hence, to obtain the maximum volume, the diameter and height of the cylinder should be 24 cm.
google is your friend. First hit:
http://jwilson.coe.uga.edu/emt725/Cylinder/cyl.html