A sphere,cylinder,and cone are of the same radius and same height. Find the ratio of their curved surface area.
since the height of a sphere is the diameter, the cone and cylinder have height 2r. The areas are thus
sphere: 4πr^2
cylinder: 2πr^2 + 2πrh = 6πr^2
cone: πr^2 + √5πr^2
sphere:cylinder:cone = 4:6:(1+√5)
To find the ratio of the curved surface area of a sphere, cylinder, and cone, all of the same radius and height, we need to calculate the curved surface area of each individually.
1. Sphere:
The curved surface area of a sphere is given by the formula: A = 4πr², where r is the radius of the sphere. Since the radius of the sphere is equal to the radius of the other two shapes, we can use this formula directly.
2. Cylinder:
The curved surface area of a cylinder is given by the formula: A = 2πrh, where r is the radius of the cylinder and h is its height. Since the radius and height are the same for all three shapes, we can plug in the values directly.
3. Cone:
The curved surface area of a cone is given by the formula: A = πrl, where r is the radius of the cone and l is the slant height. In this case, since the height and radius are the same for all three shapes, we can calculate the slant height using the Pythagorean theorem: l = √(r² + h²).
Now, let's calculate the ratio of their curved surface areas step by step:
1. Sphere:
The curved surface area of a sphere is given by A = 4πr².
2. Cylinder:
The curved surface area of a cylinder is given by A = 2πrh.
3. Cone:
The curved surface area of a cone is given by A = πrl, where l = √(r² + h²).
Now, substitute the values of r and h (radius and height) into these formulas to calculate the curved surface areas of each shape.
Once you have the three curved surface areas, simply divide the surface area of each shape by the surface area of the sphere (since they all have the same radius and height).
This will give you the ratio of their curved surface areas.
Note: It is essential to have the numerical values of the radius and height to find the specific ratio.