a hemisphericalbowl of diameter 14cm mounted on a bucket inform of frustrum of a cone if the base diametre of the bucket is 7cm and total hieght is 21cm calculate the total surface area
To calculate the total surface area of the hemispherical bowl and the frustum of the cone, we need to find the areas of each component individually and then sum them up.
1. Hemispherical Bowl:
The formula to calculate the surface area of a hemisphere is given by:
A_hemisphere = 2πr^2
Where "r" is the radius of the hemisphere. The radius of the hemispherical bowl is half of its diameter, so r = 14cm / 2 = 7cm.
Substituting the value of "r" into the formula, we get:
A_hemisphere = 2 * 3.14 * 7^2
Calculate the value to get the surface area of the hemispherical bowl.
2. Frustum of Cone:
To calculate the surface area of the frustum of a cone, we need to find the areas of its slant height, top circle, and bottom circle.
The slant height of the frustum can be found using the Pythagorean theorem as follows:
slant height = √(height^2 + (R - r)^2)
Where "height" is the height of the frustum (given as 21cm), and "R" and "r" are the radii of the top and bottom circles, respectively.
The radii of the top and bottom circles are half of their respective diameters, so R = 7cm / 2 = 3.5cm and r = 14cm / 2 = 7cm.
Substituting the values into the slant height formula, we get:
slant height = √(21^2 + (3.5 - 7)^2)
Calculate the value of the slant height.
The surface area of the frustum of the cone can be found using the formula:
A_frustum = π(R + r) * slant height + πR^2 + πr^2
Substituting the values of "R," "r," and "slant height," we can calculate the surface area of the frustum of the cone.
3. Total Surface Area:
Finally, calculate the total surface area by summing up the surface areas of the hemispherical bowl and the frustum of the cone:
Total Surface Area = A_hemisphere + A_frustum
Perform the calculations and add the values to find the total surface area.