Two cones are similar. The first one has a curved surface area of 18.4cm². if the radius of the second one is 2.5 times the first, find its surface area
Let's denote the radius of the first cone as r and the radius of the second cone as 2.5r.
The formula for the surface area of a cone is given by:
Surface Area = πr² + πr√(r² + h²)
Where r is the radius and h is the height of the cone.
Since the cones are similar, the ratio of their radii is the same as the ratio of their heights.
Let's assume the height of the first cone is h, then the height of the second cone is 2.5h.
Given that the curved surface area of the first cone is 18.4 cm², we can calculate its height using:
18.4 = πr√(r² + h²)
Substitute r = r, h = h, and the radius of the first cone as 2.5r:
18.4 = π(2.5r)√((2.5r)² + h²)
18.4 = 6.25πr√(6.25r² + h²)
Simplify the equation:
18.4 = 6.25πr√(6.25(r² + h²))
Solve for h:
18.4/6.25 = πr√(6.25r² + h²)
2.944 = πr√(6.25r² + h²)
Simplify further:
(2.944/πr)² = 6.25r² + h²
8.7402/r² = 6.25 + h²
h² = 8.7402/r² - 6.25
Now, let's calculate the radius of the second cone:
2.5h = 2.5√(8.7402/r² - 6.25)
And finally, we can calculate the surface area of the second cone:
Surface Area(2nd cone) = π(2.5r)² + π(2.5r)√((2.5r)² + 2.5h)