Final answer:
To find the surface area of a cone that is three times as large, we can use the formula A = πr(r + l) and consider the relationship between the dimensions of the cones. By multiplying the dimensions of the first cone by the scale factor, we can find the dimensions of the second cone. The surface area of the second cone is 54 x 9π.
Step-by-step explanation:
To find the surface area of a cone that is three times as large, we need to consider the relationship between the surface area and the dimensions of the cone. The surface area of a cone is given by the formula A = πr(r + l), where A is the surface area, r is the radius of the base, and l is the slant height. Since the two cones are similar, their corresponding dimensions are proportional.
If the surface area of the first cone is 54 square inches, we can set up the following equation: 54 = πr(r + l). To find the dimensions of the second cone, we can multiply the dimensions of the first cone by the scale factor, which is 3. So the radius of the second cone is 3r and the slant height is 3l. Substituting these values into the equation, we get: 54 = π(3r)(3r + 3l). Simplifying the equation, we find that the surface area of the second cone is 54 x 9π.