We can use the fact that the ratio of the corresponding lengths in two similar shapes is equal to the ratio of their corresponding surface areas. Since we are looking for the surface area of a similar cone with radius 9 inches, we need to find the ratio of the surface area of that cone to the surface area of the original cone with radius 3 inches.
The surface area of a cone is given by:
surface area = πr(r + l)
where r is the radius and l is the slant height.
For the original cone with radius 3 inches and slant height 12 inches, we have:
surface area of original cone = π(3)(3 + 12) = 45π
For the similar cone with radius 9 inches, we can find the slant height using the fact that the slant height and radius are proportional in similar cones. Specifically, the ratio of the slant heights is equal to the ratio of the radii. Therefore:
slant height of similar cone = (9/3)(12) = 36 inches
Using this slant height and the radius of 9 inches, we can find the surface area of the similar cone:
surface area of similar cone = π(9)(9 + 36) = 405π
Finally, we can find the ratio of the surface areas:
ratio of surface areas = surface area of similar cone / surface area of original cone
= (405π) / (45π)
= 9
Therefore, the surface area of the similar cone is 9 times the surface area of the original cone. The exact surface area of the similar cone is 9 times the surface area of the original cone, or:
9 × 45π = 405π square inches