If a cone of radius r and slant height 2r is surmounted on a cone having radius 2r and slant height 4r then find total surface area of solid
wondering if the lowest cone is truncated?
62.8CM
To find the total surface area of the solid, we need to calculate the surface areas of both cones and add them together.
Let's denote the first cone as Cone 1 and the second cone as Cone 2.
Surface area of Cone 1:
The slant height of Cone 1 is equal to the radius of Cone 1, which is r. The slant height is given by the formula s = √(r^2 + h^2), where h is the height of Cone 1.
We have s = 2r and r = h, so we can rewrite the formula as 2r = √(r^2 + r^2).
Squaring both sides, we have 4r^2 = r^2 + r^2.
Simplifying, we get 4r^2 = 2r^2.
Dividing both sides by 2, we have 2r^2 = r^2.
Subtracting r^2 from both sides, we have r^2 = 0.
Since r^2 cannot be equal to 0, there is no valid solution for Cone 1. Therefore, the surface area of Cone 1 is 0.
Surface area of Cone 2:
The slant height of Cone 2 is given as 4r, which is also equal to the height of Cone 2. The radius of Cone 2 is 2r.
The surface area of Cone 2 can be calculated using the formula A = πrS, where A is the surface area, r is the radius, and S is the slant height.
Plugging in the values, we have A = π(2r)(4r) = 8πr^2.
Now, we can calculate the total surface area:
Total surface area = Surface area of Cone 1 + Surface area of Cone 2
= 0 + 8πr^2
= 8πr^2.
Therefore, the total surface area of the solid is 8πr^2.