Question

Two paper cones are mathematically similar. The capacity of the larger cone is 8 times the capacity of the smaller cone. The surface area of the smaller cone is 50 cm². Find the surface area of the larger cone.​

Answer 1

200 cm²

Explanation:

The scale factor for areas of similar figures is the square of the scale factor relating their linear dimensions. For volumes of similar figures, the scale factor is the cube of the linear dimension scale factor. Together, these relations mean the area scale factor is the 2/3 power of the area scale factor.

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The larger surface area is ...

(50 cm²) × 8^(2/3) = (50 cm²) ×4 = 200 cm²

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Additional comment

The linear dimension scale factor is the cube root, or 1/3 power, of the volume scale factor. The area scale factor is the square of that or the 1/3 power times 2, the 2/3 power.

sf = ∛(v2/v1) = (v2/v1)^(1/3)

a2/a1 = sf^2 = ((v2/v1)^(1/3))^2 = (v2/v1)^(2/3)