The statement that the lateral surface area of cone A is exactly equal to the lateral surface area of cylinder B is false. Let’s break down why this is the case by examining the formulas for the lateral surface areas of both shapes.
The lateral surface area of a cone is given by the formula:
- Lateral Surface Area of a Cone:
LSA_{cone} = rac{1}{2} imes 2 ext{π}r imes l = ext{π}rl
where r is the radius of the base of the cone and l is the slant height of the cone.
On the other hand, the lateral surface area of a cylinder is given by:
- Lateral Surface Area of a Cylinder:
LSA_{cylinder} = 2 ext{π}r imes h
where r is the radius of the base of the cylinder and h is the height of the cylinder.
To compare the two formulas directly, consider that a cylinder has a constant height, while a cone tapers to a single point. This structural difference impacts their surface areas significantly.
In summary, unless specific dimensions are given that create a unique scenario where the lateral surface areas are equal (which is generally unlikely and requires exact ratios of height and slant height), the lateral surface area of a cone and a cylinder will not be the same in general. Therefore, we can confidently say that the lateral surface area of cone A is not equal to the lateral surface area of cylinder B.