a model of a solid structure in the shape of frustum of a cone with hemispherical top. The diameter of the hemispherical part is 70 cm and is equal to the diameter of the top of the frustum. The frustum has a base diameter of 28 cm and slant height of 60 cm.)
(a) The area of hemispherical surface.
b) The total surface area of the model
2/5 of the height of the cone is missing. The complete cone would have a slant height of 100.
The lateral area of a cone is πrs, so what we have is π(35)(100) - π(14)(40) = 2940π
The hemisphere has area 2πr^2 = 2π(35^2) = 2450π
The bottom of the shape is a circle of radius 14, with area 196π
So, add 'em all up for the total area.
the total surface area of the model
a) To find the area of the hemispherical surface, we need to find the radius of the hemisphere and then use the formula for the surface area of a sphere.
The diameter of the hemisphere is given as 70 cm, so the radius will be half of that: r = 70/2 = 35 cm.
The formula for the surface area of a sphere is: A = 4πr^2.
Plugging in the value for the radius, we have: A = 4π(35)^2 = 4π(1225) = 4900π cm^2.
b) The total surface area of the model includes the surface area of the frustum as well as the hemispherical surface.
The formula for the surface area of the frustum of a cone is: A = π(r1 + r2)l, where r1 and r2 are the radii of the top and bottom bases of the frustum, and l is the slant height.
The diameter of the top base is given as 70 cm, so the radius will be half of that: r1 = 70/2 = 35 cm.
The diameter of the bottom base is given as 28 cm, so the radius will be half of that: r2 = 28/2 = 14 cm.
The slant height is given as 60 cm.
Plugging in the values, we have: A = π(35 + 14)(60) = π(49)(60) = 2940π cm^2.
To find the total surface area, we add the area of the frustum to the area of the hemispherical surface: Total Surface Area = A_frustum + A_hemisphere = 2940π + 4900π = 7840π cm^2.
To find the area of the hemispherical surface (a), we first need to calculate the radius of the hemispherical part.
We know that the diameter of the hemispherical part is 70 cm, so the radius can be calculated by dividing the diameter by 2:
Radius (r) = Diameter / 2 = 70 cm / 2 = 35 cm
The formula to calculate the surface area of a hemisphere is:
Surface Area of Hemisphere (A) = 2πr^2
Plugging in the value of the radius we just calculated, we get:
A = 2π(35 cm)^2
= 2π(1225 cm^2)
= 2450π cm^2
So the area of the hemispherical surface is 2450π cm^2.
To find the total surface area of the model (b), we need to calculate the lateral surface area of the frustum of the cone and add it to the area of the hemispherical surface.
The formula for the lateral surface area of a frustum of a cone is:
Lateral Surface Area of Frustum of a Cone (A_f) = π(R_1 + R_2)l
Where R_1 and R_2 are the radii of the top and bottom circles of the frustum, and l is the slant height.
We know that the radius of the top circle (R_1) is equal to the radius of the hemispherical part (35 cm), and the radius of the bottom circle (R_2) is half the base diameter of the frustum (28 cm / 2 = 14 cm). The slant height (l) is given as 60 cm.
Plugging in these values into the formula, we get:
A_f = π(35 cm + 14 cm)(60 cm)
Calculating this, we get:
A_f = 49π(60 cm)
= 2940π cm^2
Now, to find the total surface area (A_total), we need to add the area of the hemispherical surface (2450π cm^2) and the lateral surface area of the frustum (2940π cm^2):
A_total = A + A_f
= 2450π cm^2 + 2940π cm^2
= (2450 + 2940)π cm^2
= 5390π cm^2
So, the total surface area of the model is 5390π cm^2.