To calculate the total surface area of a modified cylinder with two conical cavities, one must find the lateral surface area of the original cylinder, subtract the base areas of the cones, and then add the lateral surface area of both cones.
Finding the Total Surface Area of a Modified Cylinder
In Cylindria, we have a magical artifact that is a solid cylinder from which two conical cavities have been hollowed out at its ends. First, we calculate the surface area of the cylinder before the modification. The lateral surface area of the cylinder (without the bases) is given by the formula A = 2πrh, which is the perimeter of the base circle (2πr) multiplied by the height (h). For our cylinder, this is 2 * 3.14 * 15 cm * 70 cm.
Next, we find the surface area of the two circular bases of the cylinder, which is A = πr² for each base. However, these areas are no longer part of the surface after hollowing out the conical cavities, so these areas are subtracted.
The surface area of the conical cavities includes just the lateral surface area as the bases are now open. The lateral surface area of a cone is πrl, with l being the slant height, which can be found using the Pythagorean theorem (l = √(r² + h²)). For our cones, π * 6 cm * slant height needs to be calculated for each.
Finally, the total surface area of the artifact is the original cylinder's lateral area, minus the base areas of the two cones, plus the lateral surface areas of the two cones. Remember to calculate these values twice because there are two conical cavities.
Be mindful that the mystical standards of Cylindria dictate the use of 3.14 for π and that due care must be taken when measuring to ensure that your calculations convey the strength and resilience of this magical artifact.
The probable question may be:
In the enchanted land of Cylindria, a magical craftsman hollowed out two conical pockets from a solid cylinder to create a mystical artifact. The cylinder, standing tall at 70 cm with a radius of 15 cm, now boasts two conical hollows at its ends. Each conical cavity has a radius of 6 cm and a height of 8 cm.
What is the total surface area of this enchanted object that remains after the mystical crafting? Imagine you are an apprentice in the magical crafts academy, and you are tasked with uncovering the secret behind the surface area of this unique creation.
Additional Information:
The solid cylinder symbolizes the strength and resilience of the magical artifact, while the two conical cavities represent the mystical chambers that add an element of wonder and surprise. As you embark on this magical quest to find the total surface area, remember that in the enchanted realm of Cylindria, the value of π is considered to be 3.14 for all mystical calculations.